Given that 𝑍 is the midpoint of the line 𝐷𝐶, the perimeter of the triangle 𝐴𝐷𝐶 is 33 centimeters, 𝐴𝐷 equals seven centimeters, and 𝑍𝐶 equals five centimeters, find the length of the line 𝐴𝑌.
Well, first of all, what we’re gonna do is mark on some of the information we’ve got. So we know that 𝑍𝐶 is five centimeters and 𝐴𝐷 is seven centimeters. Well, next, if we take a look, what we can see is that 𝐴𝐷 is parallel to 𝑌𝑍, which means actually we’ve got some properties now that we can look out. And we can split up our triangle into two triangles, the triangle 𝐶𝑌𝑍 and the triangle 𝐴𝐶𝐷.
Well, because we’ve got two parallel lines and then some lines that transverse them, we can see that we’ve got some corresponding angles. So we’ve got corresponding angles here at 𝐴 and 𝑌, and then we’ve got corresponding angles at 𝑍 and 𝐷. And we can also see that we’ve got a shared angle at 𝐶. So therefore, what we can see is that in fact we have three corresponding angles between our triangles. Angle 𝐶 is equal to angle 𝐶, angle 𝑌 is equal to angle 𝐴, angle 𝑍 is equal to angle 𝐷. So therefore, we have the proof angle-angle-angle. The triangles 𝐶𝑌𝑍 and triangle 𝐴𝐷𝑌 are in fact similar. So therefore, one is an enlargement of the other. And we can also say that because of this, corresponding sides are going to be proportional.
Okay, great, so now this is gonna help us to solve the problem. Now before we use the triangle similarity to find the length of the line segment 𝐴𝑌, what we’re gonna do is mark on any other values we can using other information. Well, first of all, we know that 𝑍𝐷 is gonna be equal to five centimeters. And that’s because we’re told that both the sections 𝑍𝐶 and 𝑍𝐷 are the same. Well, we also know that the perimeter of triangle 𝐴𝐷𝐶 is 33 centimeters. So therefore, 𝐴𝐶 is going to be equal to 33 minus seven minus five minus five. And that’s because it’s the total perimeter minus the length of the other two sides of that triangle, which is gonna be equal to 16 centimeters.
Okay, great, so now what is it that we actually want to find? Well, we won’t find the length of the line segment 𝐴𝑌. But to do that, what we need to do first of all is find out a scale factor. Well, the scale factor enlargement between our two triangles, so from the smaller triangle to the larger triangle, is in fact going to be two. And that’s because if we take a look at this side here, we’re told that 𝐷𝐶 is equal to two 𝐷𝑍 because we’re told that 𝑍𝐶 and 𝑍𝐷 are in fact the same. Therefore, we know that 𝐴𝐶 is equal to two 𝐶𝑌. And we can write it as two 𝐶𝑌 is equal to 𝐴𝐶. Therefore, 𝐶𝑌 is gonna be equal to 𝐴𝐶 over two, which makes sense because if we think it’s twice as large, then 𝐶𝑌 is gonna be half the length of it.
So Therefore, 𝐶𝑌 is gonna be equal to 16 over two, which is equal to eight centimeters. Okay, great, so now we’ve got 𝐶𝑌, let’s find 𝐴𝑌 because that’s what we’re looking for. Well, the line segment 𝐴𝑌 is going to be to 𝐴𝐶, cause that’s the total line, minus 𝐶𝑌, which is gonna be equal to 16 minus eight, which gives us the answer eight centimeters. And this is in fact what we would expect. And that’s because we knew that 𝐴𝐶 was twice 𝐶𝑌. So we’d expect the line segment 𝐴𝑌 to be the same as 𝐶𝑌 so both are eight centimeters.