# Video: Finding the Perimeter of a Triangle

Given that π is the midpoint of the line π·πΆ, the perimeter of β³π΄π·πΆ is 33 cm, π΄π· = 7 cm, and ππΆ = 5 cm, find the length of the line π΄π.

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### Video Transcript

Given that π is the midpoint of the line π·πΆ, the perimeter of the triangle π΄π·πΆ is 33 centimeters, π΄π· equals seven centimeters, and ππΆ equals five centimeters, find the length of the line π΄π.

Well, first of all, what weβre gonna do is mark on some of the information weβve got. So we know that ππΆ 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 ππ, which means actually weβve got some properties now that we can look out. And we can split up our triangle into two triangles, the triangle πΆππ and the triangle π΄πΆπ·.

Well, because weβve got two parallel lines and then some lines that transverse them, we can see that weβve got some corresponding angles. So weβve got corresponding angles here at π΄ and π, and then weβve got corresponding angles at π and π·. And we can also see that weβve got a shared angle at πΆ. So therefore, what we can see is that in fact we have three corresponding angles between our triangles. Angle πΆ is equal to angle πΆ, 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 πΆππ 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 weβre gonna do is mark on any other values we can using other information. Well, first of all, we know that ππ· is gonna be equal to five centimeters. And thatβs because weβre told that both the sections ππΆ and ππ· are the same. Well, we also know that the perimeter of triangle π΄π·πΆ is 33 centimeters. So therefore, π΄πΆ is going to be equal to 33 minus seven minus five minus five. And thatβs because itβ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 wonβt 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 thatβs because if we take a look at this side here, weβre told that π·πΆ is equal to two π·π because weβre told that ππΆ and ππ· are in fact the same. Therefore, we know that π΄πΆ is equal to two πΆπ. And we can write it as two πΆπ is equal to π΄πΆ. Therefore, πΆπ is gonna be equal to π΄πΆ over two, which makes sense because if we think itβs twice as large, then πΆπ is gonna be half the length of it.

So Therefore, πΆπ is gonna be equal to 16 over two, which is equal to eight centimeters. Okay, great, so now weβve got πΆπ, letβs find π΄π because thatβs what weβre looking for. Well, the line segment π΄π is going to be to π΄πΆ, cause thatβs the total line, minus πΆπ, 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 thatβs because we knew that π΄πΆ was twice πΆπ. So weβd expect the line segment π΄π to be the same as πΆπ so both are eight centimeters.