### Video Transcript

Given that π is the midpoint of the line π·πΆ, the perimeter of the triangle π΄π·πΆ is 33 centimeters, π΄π· equals seven centimeters, and ππΆ equals five centimeters, find the length of the line π΄π.

Well, first of all, what weβre gonna do is mark on some of the information weβve got. So we know that ππΆ is five centimeters and π΄π· is seven centimeters. Well, next, if we take a look, what we can see is that π΄π· is parallel to ππ, which means actually weβve got some properties now that we can look out. And we can split up our triangle into two triangles, the triangle πΆππ and the triangle π΄πΆπ·.

Well, because weβve got two parallel lines and then some lines that transverse them, we can see that weβve got some corresponding angles. So weβve got corresponding angles here at π΄ and π, and then weβve got corresponding angles at π and π·. And we can also see that weβve got a shared angle at πΆ. So therefore, what we can see is that in fact we have three corresponding angles between our triangles. Angle πΆ is equal to angle πΆ, angle π is equal to angle π΄, angle π is equal to angle π·. So therefore, we have the proof angle-angle-angle. The triangles πΆππ and triangle π΄π·π 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 π΄π, what weβre gonna do is mark on any other values we can using other information. Well, first of all, we know that ππ· is gonna be equal to five centimeters. And thatβs because weβre told that both the sections ππΆ and ππ· are the same. Well, we also know that the perimeter of triangle π΄π·πΆ is 33 centimeters. So therefore, π΄πΆ is going to be equal to 33 minus seven minus five minus five. And thatβs because itβs 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 wonβt find the length of the line segment π΄π. 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 thatβs because if we take a look at this side here, weβre told that π·πΆ is equal to two π·π because weβre told that ππΆ and ππ· are in fact the same. Therefore, we know that π΄πΆ is equal to two πΆπ. And we can write it as two πΆπ is equal to π΄πΆ. Therefore, πΆπ is gonna be equal to π΄πΆ over two, which makes sense because if we think itβs twice as large, then πΆπ is gonna be half the length of it.

So Therefore, πΆπ is gonna be equal to 16 over two, which is equal to eight centimeters. Okay, great, so now weβve got πΆπ, letβs find π΄π because thatβs what weβre looking for. Well, the line segment π΄π is going to be to π΄πΆ, cause thatβs the total line, minus πΆπ, 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 thatβs because we knew that π΄πΆ was twice πΆπ. So weβd expect the line segment π΄π to be the same as πΆπ so both are eight centimeters.