Junior Cycle Maths - Coordinate Geometry - #SaturdaySession

0:01 hi everybody Welcome to a Saturday session with exam revision my name is Kate and today we're going to be going through the topic of coordinate geometry for higher level Junior cycle maths so firstly we'll just take a look at our learning intentions so by the end of the session what should you be able to do so you should be able to find the distance between two points the slope of a line the midpoint of two points and the equation of a line we should also be able to draw the graph of the line when you're given an equation we should be able to find the X and Y intercepts and then we all need to be able to find the point of intersection of two lines we're going to do this using two methods so number one we'll do it using a graph and number two we'll do it algebraically so that's going to be our simultaneous equations from algebra so I'm going to divide the session up

0:47 into a few sections so we're going to start at looking at the formulas where to find them and how to use them okay then we're going to look at all the information we can get from the equation of a line we're going to look at that section in a little bit more detail because it's a little bit trickier then we're going to graph a few lines and we're going to revise simultaneous equations so we use those in coordinate geometry to find the point intersection but they can also come up as an algebra question okay so it's really important we know our simultaneous equations very well okay and then we'll finish off by looking at an exam question on coordinate geometry at the end so this page is super super important it's page 18 in your formula book so you'll be given this in your exam okay so the second you see a coordinate geometry question on the paper you need to open up this page it is all

1:33 the formulas you need so you don't need to learn off any formula for this chapter okay so if the slope the length the midpoint and the equation of a line the bottom two down here we don't actually use a junior cycle mats so you can ignore them you'll see them in leaving cycle leaving certificate maths but we don't need to worry about that for now okay you'll notice here as well for the equation of the line that there's two um two equations there you need to know when it's easier to use which one okay so when we're going through the equation of the line um later on in the session we'll point out the difference between the two of them so just the first thing we need to be able to do is we need to be able to plot points so this question says plot the points one two five seven and minus one minus three below on the X Y plane the question could also say the Cartesian plane don't be thrown by the big word

2:20 it's just a normal x and y axis so with these you're always given two numbers so a point it always has a bracket and then two numbers separated by a comma so the first number tells you how far right or left to go and the second number tells you how for up or down to go how I remember this it's always when you go into the school you go along the corridor and then you go up or down the stairs okay so for the point one two we're going to go along the corridor so we'll go over one and we'll go up two okay so the point one two won't be here then five seven will be similar so you go five over this way and you're gonna go Seven Up so we are going to end up five seven we'll be there okay and then for minus one minus three we're just going to go the other way so

3:06 minus one is over there to the left and then minus three is down here okay I'm just going to add on two more examples so I'm going to say the point a 0 5. plot the point a okay so for zero five we're not going to go rise or left okay so our first number tells us how far right you're left to go we're not going to go right or left at all we're just going to go straight up to five and we'll label that as point a okay and then we'll just do point B which we'll say is minus three zero so minus 3 tells us we need to go this way and then zero we don't move up and down at all so that's the point B right there okay so points are usually given capital letter capital letters like that A and B there and this could

3:51 often come up as kind of the first part of the question so you could be told to plot the points 0 5 and -3 0. or you could be asked the opposite so you could be showing the point a here and you could be asked to write down the coordinates so next we're going to find the slope of a line okay now there's two methods for this we're going to show both methods sometimes you could you could have the choice you can pick your favorite method sometimes you need to be able to do one method over the other okay so my advice would be to be able to do both methods so we want to find the slope of the line a b and over two points here a is the point two three and B is the point five four okay first method we're going to use the formula okay so we look in our formula book and we say that okay the slope is

4:37 the form of the slope is m is equal to Y2 over or minus y1 all over X2 minus X1 okay really important that we know another way that they could write slope would be M okay so they could just use the letter n no we're just going to label our Point same as the last two methods X and y1 X2 Y2 so m is equal to four minus three all over X2 which is five minus two now I'm just going to put it all that into my calculator okay so I'm going to press the fraction button first I'm going to do 4 minus 3 all over five minus two okay so I get out 1 over okay slope is equal to one third

5:23 yeah that's method one okay method two then okay to do method two you need to have a graph in front of you okay so to do this method you need to have a graph for method one as long as you know two points you'll be able to do it okay but for method two you need a graph so instead of that formula over here on the left hand side we could write that m is equal to rise over run okay now with this just to note that formula is not in your formula book okay so you need to know that slope is equal to rise over one okay I think it's quite easy to remember anyway and it is the only we need to need to remember for this chapter okay now we're going to look at our two points we're going to look at our point

6:09 A and B what we're going to do is we're going to kind of form a triangle so we're going to draw a straight line over this way and up to there okay this here along the bottom so our left to right but it's going to be our run and then this will be our rise okay so we can see that it ran from two to five okay so how far is this distance here it's three and then it rolls up from three to four so it's one okay so it's going to fill that into our formula now so m is equal to the rise which is one all over the round which is three okay we should notice here we get out

6:55 the exact same answer so we get out one over three both methods so both methods should always give you the same answer okay just to know the second method you need to have the graph drawn and you need to learn that formula okay so now we're going to look at the first Formula so we're going to look at the distance formula so if we look at the first question here with the points a and the point B now the question says find a b so those two lines beside the letters a b that means distance okay so find the distance between A and B you need to know that notation they may not use the word distance so you need to know those two lines mean the distance between a and b now it's coordinate geometry question so we're always going to do the same thing we're going to go to our formula book

7:41 we're going to open page 18 and we're going to look for the correct formula now the word distance isn't mentioned on mentioned on that page the word length is okay and we know from everyday life length and distance mean the same thing so with all of these questions you're going to notice a pattern we're always going to write down the formula first so our formula is all right our formula is this is a big square root and underneath it it has X2 minus X1 R2 squared plus Y2 minus y1 or to be squared okay so that's our formula now we take our two points we for point A and B and we're going to label this as X1 y1 and this is X2

8:26 Y2 okay so for all our things so for distance slope midpoint we're going to do exactly the same thing we're going to write down our formula and we're going to label our points now all we need to do is solve our numbers into our formula so we've got the distance a b is equal to so you're going to have X2 which is 10. minus X1 which is 2. well to be squared plus Y2 which is 8 minus y1 which is five or to be squared I'm going to put all that into my calculator exactly the way I have it there so things like the bracket is really important 10 minus 2 3 squared plus 8 minus 5 all to be squared that is going to give me an answer of the square root of 73.

9:12 okay now when you get an answer like that again it won't always come out to be nice numbers sometimes they might okay here it didn't now it depends on what the question says so if it says round two decimal places then I press the SD button on my calculator around it if it doesn't you're better off to leave it as the square root of 73. okay now the next one same question except for there's some minor signs thrown in okay this is how you get cut out with these questions minus signs forgetting to add in the extra minus sign okay so find square root are find the distance between A and B again the same question first thing we're going to write down our formula is 2 minus X1 won't be squared plus Y2 minus y1 want to be squared and we're going to label our points so X1 y1 X2

9:59 Y2 so the distance between A and B is equal to okay X2 which is -2 then we have a minus from the formula and we our X1 is minus three so it's minus minus three I want to be squared plus Y2 which is minus one minus six all three squared okay now this is the key bit here okay so our formula is X2 minus X1 so we have minus two minus and then we can see that X1 is minus three so we need to have minus minus three now putting this into your calculator again exactly the same thing okay make sure you have all the brackets if you get an

10:45 error you have probably missed a bracket okay so like for example there needs to be a double bracket here that's minus one minus minus squared I'm getting out five root two okay again we don't need nice numbers okay numbers won't always come out nicely in maths five or two is a fine answer okay so don't think you're wrong just because you're not getting out the answer 10 or 12 or any nice number okay that's finding the distance between two points so our next Formula is for finding the midpoint okay so the midpoint we're looking for the midpoint of a which is minus 2 0 and B which is 6 8. now I'm just going to draw a very rough diagram here so 5 minus two zero and we're just

11:31 going to say it's there okay and we have six eight which is somewhere up here okay there'd be a straight line between them okay and our midpoint is right there in the middle okay so what we're looking for is a number comma another number okay we need to find out what those two numbers are okay now conveniently again our formula is in our formula book so same thing we're going to write down our formula so it's X1 plus X2 all 2 divided by two and we have y1 plus Y2 or 3 divided by 2. so we'll read our question now so find the midpoint of minus 2 0 and 6 8. we're going to label this as our X1 y1 and our X2 Y2 okay so we should be seeing the

12:19 similarity with the last section on finding the distance so now we're just going to sub this in so we'll have X1 which is minus 2 plus X2 which is six all over two and Y 1 which is zero plus eight all over 2. now when you're putting these into your calculator I'm going to put this bit in first and then this bit okay completely separate so press your fraction button first then minus two plus six then press your down button to go to the bottom of the fraction put in your two this one all works out to be two and then the same thing press the AC button the delete button to start again press the fraction button zero plus age all over two will give me four okay so that's the midpoint that is the point there right in between those two points exactly in the middle

13:06 and just one final note about slopes so we know from everyday English slope is a measure of how steep something is okay so if your line is going like this okay so if it's going uphill we expect it to have a positive answer okay or a positive slope okay so this line is going up our answer was one third okay that's positive number so that's kind of an indication that we might be correct okay if your line goes downhill okay you're expecting a negative might be minus one third it might be minus three okay but that's if your slope is going downwards when you're leading when you're reading from left to right now I mentioned at the start when we were just looking at the formulas in the

13:52 formula book that there are two formulas for the equation of the line written down there okay now why are there two formulas so first we're going to look up here okay so this formula here okay is in your formula book okay Y is equal to MX plus C now we should know already from what we just said about the slope then m is sometimes used instead of the word slope in maths okay so the letter if your equation looks exactly like this okay the number in front of the X will be your slope so if we look down here at this example okay we notice that this equation here looks exactly like the formula okay so straight away I can tell you that the slope of this line 4. okay where did I get that it's the number in front of the X

14:37 okay another piece of information I can get from it is our y-intercept okay we'll be looking at this in a bit more detail in a second but our y-intercept okay so this is my X and Y axis our y-intercept is the point where a line crosses the y-axis okay so this point here would be my y-intercept now our y-intercept all you do is you take the number on its own and you put a zero in front of it so y-intercept is the point zero seven okay now I want you to remember how to do that because in a minute when we're drawing graphs okay that's important that is one point in a line that we can find super easy okay so when our equation looks exactly like Y is equal to MX plus C we can do that now just one important thing so if

15:23 we have an equation 3y is equal to I read three Y is equal to 4C or 4 x sorry for x minus 10. okay now we can see that that doesn't look exactly like that okay now it doesn't mean we can't use it it just means we need to rearrange it a bit okay the problem is we have a three stock in front of the Y whereas the formula in your formula book doesn't have any number there okay so to get rid of that three we need to divide everything by three okay so this is in going back to algebra if you're a little bit Rusty on your algebra okay this might be a little bit difficult but when you revise your algebra manipulation formula and then you see this you should be okay so to get rid of that three we're going to divide everything by three so Y is equal

16:09 to 4 over 3x minus 10. over 3. okay now it matches up because we have our y equal to a number in front of the X plus or minus something okay so here the slope m is equal to four over three and our y-intercept would be zero minus ten over three okay just be super careful you can only read the information directly from that if the equation looks exactly like Y is equal to some number then your X and then plus or minus some number now a second way that sometimes the equations can be given is in this type of form okay no this isn't in your formula book

16:56 okay but this down here 5x plus 2y plus six is an example of another way it can be written okay there's a little trick okay again this isn't in your formula books you would have to learn it if you want to use it your slope is equal to minus a over B okay what's our a or a is our five and our B is two so slope is equal to minus five all over 2. okay for this example okay but again that isn't in your formula book if you want to learn it um then it's good it's handy when your equations are written like that okay it's kind of a little bit of a shortcut if you didn't learn this okay you could always rearrange this line okay use your algebra to rearrange it so it looks like

17:42 Y is equal to MX plus C okay so you'd get rid of this 5 from this side and you get rid of the six and then you divide them both by two okay so if you want to check that you can rearrange things it would be a good idea to try that okay but over here this is just a little trick it might speed things up a little bit if you are given an equation that looks like um ax plus b y plus c okay now a very common question is find the equation of a line okay now if you're asked to find the equation of the line okay no I've written here you must use you there is another method but for me this way is way easier so if you learn that when it says find the equation of the line that is your go-to formula okay it just makes your life a little easier um in my opinion there are

18:27 other ways to do it but I prefer to do it this way okay so the question then find the equation of the line which passes through the points four minus six and eight eight now before I even start the question the second I see find the equation of the line okay I pause and I'm going to draw out a little box so first thing I'm going to do is going to write down this equation Y minus y1 is equal to M bracket x minus X1 and I'm going to write down point and I'm going to write down okay so every single time the question says find an equation of a line it's going to be more important when we get to the harder examples later okay anytime I see it I write down that box I'm going to fill in my information so we know the points we've got the points

19:14 four minus six and I've got the point eight eight okay now I also need to know a slope okay in this question they don't tell me the slope so I'm going to go away and find it okay can I do that yes I can I can go to my formula book and I can find my formula for slope okay so we're going to go off to the side here write down the formula Y2 minus y1 1 over X2 minus X1 now I need to label my points so I'm going to call this X1 y1 and this X2 Y2 so I'm going to fill that in quickly so it'll be 8 minus minus six be very careful there you see for a minus in the formula and we're stopping in a minus six over eight minus four

20:00 into my calculator quickly so English so I'm going to get out seven over two okay you can leave it as a fraction you put in a decimal four would be 3.5 if you wish okay um whichever you prefer so the slope of our line I can say m is equal to 7 all over 2. okay now if my box fill it filled in I can go straight to my formula so we know our formula is y now this y here and this x they don't change okay so we're going to leave those as y as an X's so it's y minus y1 I'm going to go to my formula here so if minus minus six but is equal to M which just found out to be

20:46 7 over 2. bracket X that excellent change minus X1 which is minus four now we're nearly there so we're just going to have y minus minus x two minuses together need to know two minuses make a positive it's going to be y plus six is equal to seven over two x and then I'm going to multiply 7 over 2 by 4. okay sorry by minus four that's going to give me out minus fourteen okay now the question might say how they want your answer arranged okay so it might say in the form Y is equal to MX plus C okay all that means is I want this y on its own I need to get rid of

21:33 that six so let's get rid of that positive 6 we're going to subtract 6 from both sides so we're going to end up with Y is equal to 7 over 2X minus 14. minus six so we'll have y is equal to 7 over 2x minus 20. okay and that is our answer for the equation of lie okay if you had these swapped around it wouldn't make any difference okay so if you'd said that the 88 was your X1 y1 and the other one is your X2 Y2 that'd be fine sometimes your answers might come out looking a little different but they are exactly the same thing okay so there's a lot of information there for the equation of the line we are going to see another example now in a minute when we do a full question

22:20 now the next section we're going to kind of kill two birds of the one stone here so we're gonna do two things and using that method we're going to be able to answer two parts of the question so we're going to be able to graph a line and we're also going to be able to find our x and y intercepts okay so first we'll just look at an example of identifying the X and Y intercepts so your x-intercept is where it crosses the x-axis so if I look down here that's my x-intercept okay so x intercept for this example is 2 0 okay so you go two along this way but you don't go up or down any so your Y is zero okay and then your y-intercept will be this point here so my y-intercept where it crosses the y axis will be zero four okay we didn't go

23:06 right or left anywhere we just went straight up four so that's where the zero comes from now the next question you could be asked you could be given the equation of the line and you could be asked to find the X and y-intercepts okay we're also going to use this method if we're asked to graph a line okay we'll come to that in a second first we'll just pretend the question says find your X and your y intercepts okay now there's two ways to go about this you can either learn off these two sentences so to find the x-intercept you sub in y is equal to zero or you can think about it logically so the x-intercept is where it crosses the x-axis okay so like here anywhere crosses the x-axis Y is going to be zero okay and similarly for the y-intercept you sub X is equal you saw me in X is

23:52 equal to zero okay so I'd recommend trying to understand this thinking about it logically but if you prefer learning things off you can just learn off those two sentences okay now let's start so or we'll find the X intercepts we're going to sub in y is equal to zero so we have our equation Y is equal to 2x minus three so we have zero is equal to 2X minus three now we want to get X on its own so this is really good revision of your algebra so we need to get rid of that minus three we want X on its own to get rid of that minus three we're going to add 3 to both sides so we're going to end up with 3 is equal to 2X no we need to get rid of that too from in front of the X so that two is being multiplied by the X to get rid of it we need to divide so we'll end up with three over two is

24:40 equal to X so X is equal to 3 over 2 or X is equal to 1.5 if you prefer okay so my x-intercept the answer is 1.5 comma zero okay now we'll find the same equation of the line we're going to solve we're going to find the y-intercept so we're going to solve in X is equal to zero so we'll have y is equal to 2 by 0. 3 so Y is equal to minus three so our y-intercept then is zero comma minus three okay so there I found my x and y intercepts don't get the confu don't get them confused so for x-intercept y is zero y-intercept X is equal to zero so my the opposite of what you might think it should be

25:25 okay now pretend the next part of that question said graph the equation okay so the equation of a line if you want to draw a line all you need is two points we found one point we found two points so we're going to go off and we're going to graph the line Y is equal to 2x minus 3. using our X and our y-intercept okay so we're going to have 0 minus three which is down here and we have 1.50 which is here now this is easier to do um obviously on paper with a ruler you'd connect those then with a straight line okay so let's move that so it joins up so that there is our line Y is equal to 2X

26:12 okay so if the question had said draw the line or graph the line go off and find your x and y intercept you'll be able to draw the line if the question said find the X and Y intercept you also know how to do that okay now we want to look at one more so 3x plus Y is equal to 9. okay and I'm going to pretend the question says graph this graph this line okay to draw a line we need two points I already know how to find two points I know how to find the x-intercept I know how to find the y-intercept so I'm going to go away and do that okay so first we're going to find x-intercept okay at x-intercept we know Y is equal to zero so we say 3x Plus 0. is equal to nine

26:58 three x is equal to 9 divide both sides by 3 x is equal to three so we have our point three comma zero okay I'm going to plot this in a different color there so 3 comma 0 would be there and we'll also do our other one so we'll also find our y-intercept so we'll have X is equal to zero this time it'll be three comma zero plus Y is equal to nine so three by zero zeros just Y is equal to nine so we have our point zero nine okay well it's going to plot that no I don't have this drawing out very well so I'm just going to continue this up here as well as I can so five

27:43 um six seven eight nine and then we have our point zero comma nine which will be up here okay and again then I'm going to connect those as best I can obviously if you're doing this in an exam or in class you to get a ruler and you connect them perfectly with a router okay so that there is our line um 3x plus y Y is equal to 9. 9 on this slide okay in this last couple of minutes what you've learned to do you've learned to graph a line and you've learned to find the X and Y intercepts okay if you can do one you

28:30 can do the other okay so it's really good really important method here if you're asked to graph a line go find your x and y intercepts if you're asked to find the X and Y intercepts you know how to do that as well okay so really important slide here so make sure you can follow these examples um go back and listen to it again if you need to but I would consider this a really important section of the chapter so now we're going to go through a question that incorporates most of the things we've learned so far so if you're feeling confident it might be a good time now to pause the video give the questions a go and then check back in to correct your work um if you're still a little bit unsure stay going with video and we'll work through the solution so this question starts off as expected so we're given two points two two and minus 4 5 and we are told from part A to

29:17 plot the points okay so plot point a which is 2 2 2 2 means that way and two up that way 0.22 is here I'm going to label that as a and then with the point minus 4 5 is B so minus four would be this way and then up to five so this will be our Point minus four five right here okay part eight on nice and straightforward like you'd expect your first part to your questions to be then Part B says find C the midpoint of a b okay any question like this first thing I'm going to do open up my formula book and I'm going to write down my formula so my Foreigner from midpoint is X1 plus X2 divided by 2 and y1 plus Y2 divided by 2. write down my points my point a is 2 2 x 1 y 1 and my point B is minus four

30:05 five X2 Y2 and I'm going to set this into my formula so we have X1 which is 2 plus X2 which is minus 4. all over 2 and we have B which is minus sorry Y2 which is five plus y1 which is 2 all over two I'm gonna put this into my calculator exactly the way it is there and I'm going to get out minus 1. and I'm going to put the second part into my calculator I'm going to get out 3.5 easier to change to a decimal here because I'm going to plot it so I'm going to plot my point C here minus 1 3.5 which will be there okay I know by just looking at that that looks correct okay because it's halfway between A and B so

30:52 it's that's my point C the midpoint but so part C says to verify that now remember from earlier that these two lines here mean the distance so the distance AC is equal to the distance BC okay really what we're going to do here we're going to divide our question into two parts so first we're going to find AC and then we're going to find BC and if we're correct our answers will work out the same so we're 400 for distance okay it's written as length in our formula book length distance at the same time so first we'll do a which is 2 2. X1 y1 and we'll do c which is the point we just found up there minus one

31:37 point five which will be X2 Y2 so we're going to solve that in so find AC is equal to minus one minus two all three squared and 2.5 minus two to be squared so put that into our calculator and we get out 3 root five over two so the distance between a and C is 3 root five over two don't be alarmed that it's not a nice number okay it's still a perfectly valid answer now we're going to find BC okay so B is the point minus four five

32:22 and C is the point minus one 3.5 so X1 y1 2 Y2 okay now before I even start I know my answer should be three root five over two so fingers crossed that's what we get out all right would be minus one minus minus four okay be careful there we're double negatives we'll be squared put that into my calculator so I get out the same thing so I know I'm correct so three root five over two so therefore those three dots mean

33:08 therefore we can say that AC is equal to see we got it the same answer for both the part D says to find the equation of a b give your answer in the form Y is equal to MX plus C now we'll worry about the last bit at the end the question says find the equation okay we're finding the equation of a line I'm going to make out my little box so anytime it says find the equation of the line write down y minus y1 is equal to M bracket x minus X1 we write down point fill in what points we know we know two two and minus 4 5. and we also know another Point as well but we don't need to know three points for this so as long as we know two that's fine and slope

33:57 okay now I actually don't know the slope yet so I need to go off and find that now there's two methods I could do I can do rise over run or I can do the formula now I actually prefer the formula but that's just personal preference um whichever you prefer so m is equal to Y2 minus y1 all over X2 X1 so m is equal to 5 minus two all over minus four minus two I'm gonna put all that into my calculator okay and I get at the slope is equal to minus okay I'm expecting that because this

34:44 line here looks something like that it's going downhill okay so we're expecting a negative slope so m is equal to minus a half I have enough information so I can just sub it into my formula remember we don't change the normal X and the normal y it's going to be y minus 2. equal to minus a half bracket x minus X1 which is 2. so y minus 2 is equal to a half X minus a half multiplied by minus two will give me plus one okay now that is my equation of the line if I go back up to the question it said in the form Y is equal to MX plus C so basically we want the Y to be on its own this minus 2 is our problem to get rid of a minus 2 we need to add 2 to the other side so to be Y is equal to

35:29 sorry I forgot the X there so minus a half X 1 plus 2 will give me three okay so that is my equation of the line in the form Y is equal to MX Plus the next thing we're going to look at is the point of intersection so the point of intersection is where two lines meet okay so if this was line L and this was line M their point of intersection is Reds dot there okay now there are two ways to find us it depends on how the question is asked what work you've already done as to how you're going to do it so method one you could graph both lines and read the point from the graph okay so remember to graph both lines find the X and Y intercept them both and read it off of it okay so for example if these were two lines you'd find the X and Y

36:15 intercept of the Top Line the X and Y intercept the bottom line draw your two lines find your point in the middle okay that's one way to do it method two then is algebra and it's using your simultaneous equations okay so we're going to quickly revise those okay so we're going to pretend that this is the line L and this is the line M your question would say find the point of intersection of l and m okay now draw your for your simultaneous equations we're going to write down our two equations so minus X plus six and Y is equal to X plus two okay for simultaneous equations we need something to cancel we can actually see here already that the X's will cancel because we have a minus X on top and a plus X on the bottom cross those out so you'll be left with

37:01 2y so we'll add the Y and the Y here is equal to 6 plus 2 is equal to eight 2 multiplied by 4 gives me eight so divide the eight by two get Y is equal to four okay that's one of my answers now to get out the X either the top or bottom one I'm going to go with the bottom one y is equal to X plus two and Sub in what you just found so Y is equal to 4 so 4 is equal to X plus two plus two is equal to four x must equal two okay now if it was an algebra question and if this came up in just a pure Algebra I just said find or solve the simultaneous equations that would be a fine way to write your answer okay but

37:47 when it comes up in coordinate geometry the question will say find the point of intersection so we need to put our Point together so our X always comes first so our point of intersection will be 2 4. okay so that's a good revision of simultaneous equations if you are a bit roster on your sometimes equations make sure you go back and revise them when you're doing this chapter okay now the second one here is just a little bit trickier so again pretend the question says find the point of intersection of these two lines now when we're writing these out okay we can immediately see that nothing is canceling straight away okay now I'm going to pick to cancel my Y's just because it's a little easier here but we need one of those to be a minus one y okay so we're going to multiply all the bottom line by minus one okay Top Line will leave alone Y is

38:34 equal to 2x plus 3. probably the biggest mistake people make here is they forget to multiply something in the line by the minus one or the minus two or whatever it is okay so here we need to multiply all the bottom line by minus one so I'll change to minus y is equal to minus five x multiplied by minus one will be a positive five x and ten multiplied by minus 1 will get minus 10. okay now these will cancel so we'll be left with zero is equal to seven x minus seven no we want X on its own we need to get rid of that minor seven to get rid of that minor seven we'll do the inverse operations we'll do the opposite so we'll add seven so this will be seven so zero plus seven will give me seven is equal to seven X and now we'll divide both sides by seven so X is equal to one

39:21 okay that's half my answer now we'll sub that answer into one of our original equations we'll go into the top one so Y is equal to two X plus three Y is equal to two by one plus three because we just found out that X is equal to one so Y is equal to five okay so then again if that was algebra that'd be your answer done when it's simultaneous equations you'll be looking for your point of intersection so your answer will be one comma five remember your X always comes first okay so what I'd say about this now I went to the simultaneous questions there quite quickly if you're unsure about those make sure you go back and revise them from algebra okay so you need to be clear about how you revise when you're revising simultaneous equations look at it from coordinate geometry point of

40:07 view as well when you're revising coordinate geometry make sure you go back to your simultaneous equations from algebra okay so that's for your point of intersection basically what you need to learn is point of intersection go straight to simultaneous equations okay unless you already have your graph drawn or it says from your graph Point find the point of intersection if it mentions anything about algebra you need to do your simultaneous equations using the algebra method like these two examples knowing what parallel and perpendicular lines mean is really important for coordinate geometry okay so we're going to be looking at these in terms of slope so first for parallel lines okay so the easiest way to think about these is train tracks okay so train tracks always run parallel okay they'll never meet

40:53 and what parallel lines mean a coordinate geometry is they have the same slope okay so we can see here if the law if the slope of line 1 here is a half the slope of this line is a half so they're parallel okay if the slope of this one here is two they're parallel because the slope of that one must be two okay I'm going to call this line L and I'm going to call this line k you might see this symbol L parallel to K so those two lines that kind of look like terrain Jacks that's the symbol for parallel okay so if they're parallel they have the same slope we're going to be seeing questions in a second where you need to use that knowledge and apply to the question but for now just know that parallel lines have the same slope okay so you need to learn that off

41:38 then the next one is perpendicular lines okay so perpendicular lines they meet at a right angle okay so if we draw a line here we'll call that line L and we're going to call this line K if they're perpendicular all these angles in here are right angles so they're 90 degrees okay now it says here perpendicular lines the slope of one is the reciprocal of the other okay so if the slope of line L is three let's say minus three over four okay so the slope of L minus three all over 4. and the question says find the slope of K slope of K so what you do is you flip it upside down

42:24 so instead of writing three over four we're going to write four over three and we're going to change the sign so it was a negative it's now a positive okay if one line had the slope of 3. okay the perpendicular slope would be minus one all over 3. okay if something at the slope of minus one over five the perpendicular slope flip it upside down so it just become five and change the sign it'll become positive okay so flip it upside down and changed the sign okay um what's really important as well just the symbol um so if L is perpendicular to K you write that as L and collect an upside down capital T so that symbol there means perpendicular

43:09 okay so you need to know perpendicular lines to get the perpendicular slope flip it upside down and change the sign parallel lines they have the same slope you don't need to do anything okay now they come open slightly trick your questions that we're going to move on to now okay so we need to know this knowledge you can learn that off by heart okay so you do need to learn certain parts of this chapter there are in formulas for everything this is one of those parts that you need to learn off okay yeah the slightly harder question for finding the equation of a line okay now just because these are harder we're going to apply the exact same things that we did with the more basic questions and then we wanted a problem so p is the line Y is equal to X plus four the line Q contains the 0.56 and is

43:55 parallel to p find the equation of Q okay now what we're going to do is we're going to write down our box why are we going to write down our box for finding the equation of the line because all we're really asking is to find the equation of Q anytime we're asked to find the equation of a line we write down our formula I'm going to write down point how do you write down smoke we're looking for the equation of Q do we know a point on Q we do it's 5 6. and do we know the slope no we don't know the slope okay now I think what methods do I know to find the slope so I know my formation is equal to Y2 minus y1 over X2 minus X1

44:42 for that we need to know two points we only know one point so that's not going to work you can also find the slope using rise over run we don't have a graph drawn that's not going to work so I need to go back to the question okay we're told in the question that P is the line Y is equal to X plus 4 and this is parallel to Q so p and Q are parallel what does that tell us they're the same slope so what do we know about P we know that P is the line Y is equal to X Plus 4. okay what does that tell us I can find my slope from that so it really looks like Y is equal to MX

45:27 plus C the only difference is there's no number in front of the X when there's no number in front of the X there's really a one there okay really people like mathematicians were just too lazy to write that one but there's nothing wrong with writing it and for this question it makes a little easier so m is our slope the number in front of the X which is one okay p and Q are parallel so this is our slope of P and it's also our slope of Q because they're parallel so we can write up here m is equal to one okay now I have all my blocks filled in so I can just go ahead and fill in the equation so this will be my X1 and my y one so y minus 6 is equal to one bracket x minus five I'm just going to multiply that I

46:13 can tie it up a little bit so y minus six is equal to X minus five okay now we need to get rid of this minus six from this side to get rid of the minus six We'll add six to both sides so Y is equal to X plus one minus five plus six will give me plus one so that is our equation of Q okay so again anytime it says find the equation of the line this box is your savior okay you really can't go wrong if you use that box okay so anytime it says find the equation of a line do that box fill out what information you can and then you may have to find the slope using a method like that so

46:59 okay so just one more similar question so here m is the line 3x minus 6y plus 11 is equal to 0. the line n contains the point minus 4 2 and is perpendicular to M find the equation of n no I've read the question this is my kind of key bit find the equation of n so I'm going to write out my box point and slope okay do I know a point on N I do it's minus 4 2. okay I'll label that as my X1 y1 do I know the slope no okay so again I know one point can't use the formula because I need two points for that can't do rise over run because I need to have a graft run so I look at what other information

47:46 we're given the equation of the line M which is 3x minus 6y plus 11 is equal to 0. okay and this is perpendicular to n so if I can find the slope of M I'll be able to flip it upside down and change the sign and that will give me the slope of n now to find the slope of M okay I'm going to do it both ways okay so I'm going to do method one okay so 3x minus 6y plus 11 is equal to zero okay that's in the form ax plus b y plus C is equal to zero okay so for this method when it's laid out like this okay

48:31 slope is equal to minus a all over B here a is the number in front of X so a is equal to Three B is the number in front of Y so B is equal to minus six okay so when I fill that in slope is equal to minus three all over minus six which is equal to okay that's method one now if you don't like that method you can do method two which is used a little bit more algebra okay we should get out the same answer so we have our equation 3x minus 6y plus 11. is equal to zero and we're going to change it into the form Y is equal to MX plus c now it's going to be a little bit of rearranging involved here so what I need to do is I need to get this 6y on its

49:17 own okay so my problems are the 3x and the 11. okay so I'm going to let minus 6y is equal to now I need to get rid of this minus 3x so what I'm going to do is I'm going to subtract minus 3x from both sides so I'm going to end up with minus 3x here and I need to get rid of this 11. we're going to subtract 11 from that side as well okay so now we're close our only problem now is this minus 6 in front of the Y so what we're going to do is we're going to divide everything on the other side by minus 6. so it'll be minus 3 over minus 6x minus 11 over minus 6. okay so now we're left at Y is equal to minus 3 over minus six it's going to give me a half X um and then that will be plus 11 over 6 because the minuses will cancel out that

50:04 is actually quite irrelevant because all we want is y is equal to MX plus c okay we can see that our m is the number in front of the X m is equal to a half okay so we got out the same answer using both methods okay obviously this is a lot shorter but you do need to learn off that the slope is minus A over B okay here you need your algebra to be quite strong because you're doing a lot of rearranging okay but all I wanted to show there was you get the same answer both ways so anyway our slope of m of m is equal to a half no need to be careful the Temptation would be just to fill in up there m is equal to a half okay but if

50:51 we're looking for the slope of n and n is perpendicular to n so slope of m a slope of M even so we're going to do to find the perpendicular slope we're going to flip it upside down so we'll end up with two over one and we're going to change the sign we're going to put a minus sign in front of it okay so really what that is is just minus two if you put this into your calculator and just tell you that it's -2 you can ignore that one underneath okay so our slope is -2 so now we have our box filled in we can go straight and fill in our formula so let's be y minus two is equal to m which is minus two x minus X1 which is minus four so y minus 2 is equal to we'll do what's inside in the bracket first so we'll get

51:37 rid of these two minuses that can be X plus four so y minus 2 is equal to minus 2X minus 8 and now let's get rid of this minus two so we'll add 2 to the other side minus 2x minus a plus 2 will give me minus 6. so this here is my equation of n okay so a little bit more work involved here obviously you don't have to do the two methods to find the slope all I did there method one and Method two I just want to show that you get out the same answer both ways okay so the key to both of those last two questions throughout your box write down what information okay always write down as well what you actually found so in this first step I found the slope of M

52:23 back up and read your question you don't want to find the equation of M we already know that we want to find the equation of n so you need to flip it upside down and change the sign because they're perpendicular so slightly harder questions but I think very doable when you have it laid out correctly and keep your work nice and organized so just to finish up we're going to take a look at two exam questions they're actually bowled from last year's Junior so 2022 so we're just going to see how coordinate geometry was asked okay um kind of the questions cover nearly everything that we've done so far okay so they they were really good questions and if you can do these questions I would say that you're fairly confident in this topic okay so again if you're feeling confident about this be a good idea to pause this video you can see at the top of the question here it says suggested time five minutes okay so take

53:10 your five minutes make sure you don't go over it see how much of it you could get done and then we will come and correct it okay so I'm gonna read the question first the line H has a slope of 4 and passes through the point 2012. find the coordinates of any other point on the line H other than the point 2012 sure you're working it okay now here what we notice okay they don't have like one two three four okay so we can't just read a point directly from the graph okay now I'm going to look at the information that I'm given I'm given a point and I'm given a slope what might be useful if I could find the equation of the line okay so can I find the equation of the line H okay I have a point I have a slope

53:55 that's exactly the information I need to find the equation of a line so maybe that might help me that's what I'm going to do I'm going to find the equation of the line okay so I have my formula okay and you should know the drill at this stage first thing we're going to write down is our point which we know is 20 12. this one y one and we also know our slope which is m is equal to four I know all my information so I was going to go ahead and fill it into my formula so y minus 12 is equal to M which is 4 bracket x minus 20. so y minus 12 is equal to 4X

54:41 minus 80. you need to get rid of this 12 from this side so to get rid of a minus 12 We'll add 12 to both sides so if the Y is equal to 4X now minus 80 plus 12. is minus 68. okay now I would always check just I know that's a fairly straightforward calculation but if you're anyway unsure your minus signs always double check that in your calculator because you don't want to be missing marks like that um on things you could have done on your calculator no that is not my question okay it's probably a lot of my 10 marks but it does not answer the question so we want to find the coordinates of another point now technically there's so many answers to this okay there's one obvious Point kind of jumping out at me here and it's that point okay another word for that point is my X

55:28 intercept I have my x-intercept we're going to find out how my x-intercept Y is equal to zero so I have my equation Y is equal to 4X minus 68 so I'm going to solve in y for zero so zero is equal to 4X minus 68. I want X on its own we need to get rid of this minus 68 so you'll add 68 to both sides so it'll be 68 is equal to 4X you need to get rid of that 4 so we're going to divide both sides by 4. 4 [Music] 17. so my x-intercept is the point 17.

56:15 0 okay so that point there should be 17 0 so that's my answer okay now that's just one possible answer okay you didn't have to find the x-intercept the only reason I chose that is we've been doing a lot of work on that in this session okay other options I'm not going to do these out but um these are all the things you could have done so instead of finding the x intercept you could have found the y-intercept okay so you would have subbed in x equal to zero and found your y okay you would have actually gotten um we just do this quickly so Y is equal to 4 by 0 minus 68. so Y is equal to

57:03 minus 68 so your y-intercept would have been 0 minus 68. that would have been another answer okay now what you've got also done you could have put in X is equal to one unsolved for y so you could have said that Y is equal to four by one minus 68 so Y is equal to 4 minus 68 like minus 64. it's that point one minus 64. would be another possible answer okay or dot dot you could have found X is equal to 2 Sub that in and found the Y okay so you actually could have done an infinite amount of things there and gotten the right answer okay the key thing is that

57:50 you found the equation of the line once you found the equation of the line you should be flying with that question okay x-intercept there was kind of the obvious one to me but again you could have found x equal to one solve for y x equal to two solve for y you could have left Y is equal to 10 and solved for X okay so you really could have done anything but I think they're the x-intercept is the most obvious choice okay so just be aware if you didn't get my answer there your answer may still be correct um so now we're moving on to a slightly longer question okay so it's apparent a b c and d of this question so we read the question first just to figure out what we're being asked so the coordinate diagram below shows part of the n22 road at County Cork okay not

58:36 very relevant to the question two points on the road p and Q are marked in the diagram okay so now we know we've two points on the diagram we know it's coordinate geometry question now part A says the point Q has coordinates 6 2 write down the coordinates of the point P so look for the coordinates here so the coordinates are minus one okay now some of you might be thinking why did they bother with this slide okay say for example you drew a complete blank in the example okay and you couldn't remember which number comes first if this is 6 2 okay how did they get the six they use this number first and then this number you could apply that knowledge then to figure out which number comes first for p okay just in case you drew a complete blank no Part B then the equation of the line PQ is X plus seven Y is equal to 20.

59:23 using this or otherwise find the coordinates of the point where PQ crosses the y axis okay now usually when it says using this or otherwise the otherwise method isn't usually a straightforward okay so just go what they said we'll use the equation so we have X Plus 7y is equal to 20. we're looking for a way across the y axis so the y-intercept okay at the y-intercept we know X is equal to zero so I'm just going to sub that in so it's going to be zero plus 7y is equal to 20. 7y is equal to 20. how will I get rid of that 7 from in front of the Y we're going to divide the 20 by 7. so Y is equal to

1:00:09 20 over 7 our Y is equal to okay now that doesn't change to two nice a decimal okay so we'll actually just leave it as the fraction so Y is equal to all right so our y-intercept would be 0 comma 20 over seven okay if we want to kind of approximate it it's actually two point eight five okay now does that look about right so if I was to draw a line connecting p and Q okay so it does cross the x axis or the y axis there around two point something so my answer doesn't make sense okay I didn't say to decimal points or anything so we can just leave it in fraction form because it's easier

1:00:55 um but if it did you'd have to know your rules around it it's part A and B though now let's look forward to part C a new road is being built through Q which is a 0.62 on the coordinate diagram which will be a straight line segment which is perpendicular to PQ okay I'm going to go back to my diagram here perpendicular to PQ okay what does that mean it's going to measure at a right angle so it would be something like this roughly okay now that's a very rough line but that would be roughly what it looks like that red line there okay I might just change the color of that so it looks different to the original line so that green line there is what we're looking for okay work out the equation of this new Road

1:01:41 okay now equation of the new Road we're looking for the equation of a line first thing I'm going to write down y minus y1 do we know a point on this we do the point is six two do we know the slope we don't okay I don't know the slope but we know how to find it what is the talent we tell it tells that it's perpendicular to PQ okay now the line PQ has equation X Plus Y X plus seven Y is equal to 20.

1:02:27 okay now can I find the slope of this okay so what I'm going to do I'm going to rearrange it to be in the form Y is equal to MX plus c okay so I'm going to get Y in its own so it'll be 7y 20 minus X okay now I'm going to rearrange that because we always have our X first seven Y is equal to minus X Plus 20. now I need to get rid of that seven so Y is equal to minus 1 over 7 x plus 20. so our slope m is equal to minus one over seven okay you could also rearrange just to be

1:03:15 in the form ax I'll do that quickly here boy equal to zero so that would give us um X plus seven y minus 20 is equal to zero slope is equal to minus A over B so there's a one in front there so slope is equal to minus one over seven you get out the same answer okay either way whichever one you prefer now that's the slope of our original road so that's the slope of this road here okay but the road we're trying to find the equation of the green one there okay it's perpendicular so what we need to do is we need to flipper

1:04:01 slope upside down and we need to change the sign okay so if we flip this upside down we're going to get seven over one I'm going to change the sign so it's going to become positive so slope really here is seven okay so this is really really similar well it's actually the exact same question as the second trickier one we did um sorry this one here okay so it's essentially the exact same question okay now it's going to fill it in so we'll have y That's my X1 y1 so it'll be y minus two is equal to seven bracket x minus so y minus 2 is equal to 7X minus 42. now this is where we're going to read our question now we haven't seen this before this a slight extension onto

1:04:47 what we've done we want our answer in the form ax plus b y plus C is equal to zero okay basically we want here we want our X's y's and our numbers all at the same side okay so we're going to put a zero well we bring everything to let's say we'll bring everything over to that side over to the left hand side so we'll have minus 7x okay so we need to get rid of the 7x from this side to get rid of that we'll subtract it from the other side plus y minus two to get rid of this 42 from this side we're going to add 42. is equal to zero so we'll be left at minus 7X plus y Plus 40. is equal to zero okay so that now is in the form something X plus something y plus

1:05:32 something is equal to zero okay so they can either ask it like that so they can either ask it in the form Y is equal to MX plus C or to be in the form ax plus b y plus C is equal to zero okay just make sure when you find the equation of your line you go back up and read the question did it give you any criteria okay because you'd be losing the last couple of marks after all now Part D actually has nothing to do with chord mixometry we're not going to do it in this video just because we haven't done any preparation for it um but if you have ratios done already in class it would be great to attempt this question as part of revision okay uh my topic tip here would be just watch your units say centimeters millimeters and meters okay

1:06:20 um and then they want their answer in kilometers let's just make sure you're converting between them correctly and also make sure that your answer makes sense so if you look at the map okay it's a map of a road in County Cork so those two points okay if you look at the river like those two points aren't super close so those two points are Antonia a couple of hundred meters apart but they're also not thousands of kilometers away from each other okay so just make sure your answer at the end there makes sense okay but you should be fairly prepared now that when you get a question and you do certain maths it's not just going to be one topic you might have to pull your knowledge from a couple of different topics into the same question okay this is a good example of it here it was all a coordinate geometry and then suddenly erase us okay so just end just some tips for how I

1:07:07 would approach this topic when you're revising so first thing always write out the formula and label your label your points so by label your points I mean your X1 y1 your X2 Y2 and when you're writing out the formula okay some people think okay I've done loads of these questions I know the formula make sure you still look at the formula book okay because you could make a small mistake with a minus sign or a plus sign or something okay so always write out the formula from the formula book when you're studying as well make sure you get familiar with using your formula book um just because the formulas might look slightly differently in your textbook then be careful of the double minus signs what I mean by that is in some formulas there's a minus sign in it and if you're subbing in a negative number the two negatives will make a positive

1:07:53 we saw some examples of that um so just be careful when that happens the second the question says find the equation of the line draw out the box so your formula y minus y1 is equal to M bracket x minus X1 and your point and your slope fill it in and then you really can't go wrong with that question if you're drawing a graph make sure to make them neat and use a ruler if you are asked to draw a line between a and b and you don't use a ruler okay you will definitely lose Max okay so make sure you use a ruler and make it nice and neat and then finally practice exam style questions so that's the only way you get used to the phrasing of the questions and also get prepared for questions that will link into other topics so for example with the simultaneous equations um when I'm studying for this I'd learn my simultaneous equations but I'll also

1:08:38 go into my algebra chapter and just revise it from an algebra point of view as well okay so I hope you found this video useful and best of luck with your sort of