The MOST DIFFICULT Grade 9 Coordinate Geometry Exam Question

hey guys what's up so today we're doing yet another uh grade nine questions so again this is actually paper one so once again there's some non-calculator paper question and it looks something like this so we have a straight line here so first of all ABCD is a big old rectangle a E B on one line where the equation is but let's write out one more time X Plus 2y equals 12. and we also know that this line here is a straight line so A and D are on a straight line M and we need to find an equation for M it also tells us that a to e and e to B are the same so the first thing we need to do is work out some stuff right because we have absolutely no information at the moment so we need to make sure that we actually get something so with an equation of a straight line what can we work out well you can always work out the X and Y intercepts which here are points e and B respectively so the way we have to do that is to actually rearrange this equation first so I'm going to subtract X on both sides so minus X plus 12 because we want it in the form y equals MX plus C so divide both sides by two and we get minus a half X plus six so for the x intercept we so actually I do the Y intercept first so the coordinates for e it's the y-intercept so what we're going to do is sub in x equals zero because remember all along the y-axis X is equal to zero and that gives me Y is equal to minus a half times zero so just zero six so for E the coordinates are 0 and 6. then for point B that is just the x intercept so we're going to sub in y equals zero so that gives me zero equals minus a half X plus six I'm going to move the minus a half x to the left hand side which gives me a half X equals six which means x equal to 12. so the coordinates there are 12 and 0. so now it's a good time to actually think about what's happening with this question so we want to find an equation for this straight line here n so if we write M right that's going to have y equals MX plus c now because it's a rectangle I know that the line aeb is perpendicular to line DM so the gradient of it is going to be the negative reciprocal of the gradient here so the gradient here is minus a half so you flip it to make it two and then you change the sign to make it positive two so we have y is equal to 2x plus c now how do I work out plus C well I can either work out what the y-intercept is but it looks like it's going to be off our actual axes or I can just find another coordinate now that lies on the line and subbing that coordinate and I can work out C so the only coordinate that I believe I have any information about is a because it intersects with my other line so here's what I'm going to do is if I look at a we should remember that the distance a to e is the same as e to B so to go from E to B I've minus 12 from the x coordinate right which means the x coordinate of a would minus another 12. okay with the Y coordin I've gone from zero to six I've gone up by six I'm going to go up by another six okay so in other words this triangle that I've made from A and E like this is the exact same as the triangle like this okay it may not look it but it is so hopefully that makes sense so I do have sorry it's not going to be six it's going to be 12. I meant it's going to add 6 which makes 12. thank God I spotted that right away so all I need to do now is sub that into this equation to work out what C is so when X is equal to -12 Y is equal to plus 12. so something in I get 12 is equal to 2 times -12 plus c that gives me 12 is equal to minus 24 plus C which means C is equal to 36 so my final equation for m is y is equal to 2X plus 36 now does it make sense well the lines going up so it should be a positive gradient 36 well I know it's off the top of here so yeah I think that's a pretty reasonable thing now if you notice it's actually four marks it's not that much work for four marks but it is pretty tricky to kind of realize so again my kind of General exam kind of approach is to always look at what I need and then work it out so for a straight line here I need to get the gradient and the Y and another coordinate right well the gradient I can work out from just the equation of this line because they're perpendicular it tells me it's a rectangle that rectangle thing has to be useful in this case we used it to show that the lines are perpendicular you can never assume it you have to use information so and then from there we can just kind of figure out what we can again with this part here a is equal to EB again we use that in order to work out the coordinates of a so I hope that makes a little bit more sense and I hope you enjoy and found it useful