The grid shows the points 𝐴 and 𝐵. Part a) State the coordinates of the point 𝐵. Part b) On the grid, mark with a cross a point 𝐶, such that 𝐴𝐵𝐶 is a right-angled isosceles triangle. Part c) On the grid, mark with a cross the point three, one. Label this point 𝐷.
So the first thing we’re interested in is point 𝐵 because we want to find the coordinates of point 𝐵. And whenever we’re trying to find the coordinates of a point, we always start with the 𝑥-coordinate. So actually, we can also say that it’s along the corridor. So we do along the corridor then up the stairs or downstairs. That’s one way of remembering to do 𝑥-coordinate, then 𝑦-coordinate.
We can see here the 𝑥-coordinate is actually at 𝐴 cause we go along from zero to eight. And we can see that actually the point 𝐵 is up from there. And then, we move on to our 𝑦-coordinate. And we can see here that the coordinate is going to be three and that’s because we go from zero up to three. And we can see that along, we’d actually see that 𝐵 is at this point. So therefore, we could say that the coordinates at the point 𝐵 are going to be eight, three.
Okay, now, let’s move on to part b. In part b, we need to draw our point 𝐶 such that 𝐴𝐵𝐶 is a right-angled isosceles triangle. So the key thing here is “what is a right-angled isosceles triangle?” Well, here, I’ve drawn a little sketch to help us understand.
So a right-angled isosceles triangle is a triangle with a right angle; seeing this arrow pointing at here. And also, it has two identical sides and we see those with these little lines here cause that’s what it means; it means these sides are the same. And then, we’d also have two angles. So our two base angles here would also be the same. So this is a right-angled isosceles triangle.
So now, let’s draw a one on our grid. Well, actually, I’m gonna show these three places where we could actually put point 𝐶 so that 𝐴𝐵𝐶 is actually a right-angled isosceles triangle. So the first where we’re gonna have a look at is to have a right angle at 𝐴.
So in order for it to be a right-angled isosceles triangle, what we need to do is have two sides which are actually the same length. So we’ve got 𝐴𝐵, which is six units. So therefore, we’ve actually placed the point 𝐶 six units away from point 𝐴. So therefore, we can say that 𝐶𝐴 is six units long or six squares.
So now, you can see that we’ve actually answered the question because we’ve got a point 𝐶 and it’s at the coordinates two, nine, where the triangle 𝐴𝐵𝐶 is a right-angled isosceles triangle. And that’s because we have two sides of the same length and a right angle.
Okay, so now, let’s look at the other two possibilities for where 𝐶 could go. Well, the next possibility is to have the right angle at point 𝐵. Again, 𝐴𝐵 is six units long. So we’ve placed the point 𝐶 six units away or six squares away from point 𝐵. So we can say that 𝐶𝐵 is six units long now. So therefore, the point two, three is another possible point for point 𝐶 because we’ve now drawn again another right-angled isosceles triangle.
Okay, so now, let’s move on to our last point. Now, for our last possible point, we actually want the right angle to be away from 𝐴 or 𝐵. And the only way for that to happen would be to have the angles at 𝐴 and 𝐵 as 45 degrees.
So here, we can use the grid squares to help us because we can actually go out from 𝐵 and 𝐴 at a diagonal. So therefore, we know it’ll be 45 degrees because we’re dealing with squares. And we can actually go three squares diagonally out. And we can actually see we get to the new point 𝐶 at five, six cause that creates a right angle. And we also have two identical sides.
So we’ve answered part b. We’ve actually drawn three points such that 𝐴𝐵𝐶 could be a right-angled isosceles triangle.
So finally, part c, what we need to do is actually mark on the grid the point three, one. Well, we already know from earlier that when we’re looking at coordinates, the first coordinate is our 𝑥-coordinate and then the second coordinate will be our 𝑦-coordinate. So we know three will be our 𝑥-coordinate and one will be our 𝑦-coordinate.
So I’ve actually marked these on our 𝑥- and 𝑦-coordinates. So we got three and one. We can see that where these points meet is going to be the point 𝐷 at the coordinates three, one. So we’ve answered part c as we’ve marked on the grid with a cross the point three, one.