Video: Finding the Perimeter of a Triangle

Given that 𝑍 is the midpoint of the line 𝐷𝐢, the perimeter of △𝐴𝐷𝐢 is 33 cm, 𝐴𝐷 = 7 cm, and 𝑍𝐢 = 5 cm, find the length of the line π΄π‘Œ.

03:45

Video Transcript

Given that 𝑍 is the midpoint of the line 𝐷𝐢, the perimeter of the triangle 𝐴𝐷𝐢 is 33 centimeters, 𝐴𝐷 equals seven centimeters, and 𝑍𝐢 equals five centimeters, find the length of the line π΄π‘Œ.

Well, first of all, what we’re gonna do is mark on some of the information we’ve got. So we know that 𝑍𝐢 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 π‘Œπ‘, 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 πΆπ‘Œπ‘ and the triangle 𝐴𝐢𝐷.

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 π‘Œ, and then we’ve got corresponding angles at 𝑍 and 𝐷. And we can also see that we’ve got a shared angle at 𝐢. So therefore, what we can see is that in fact we have three corresponding angles between our triangles. Angle 𝐢 is equal to angle 𝐢, angle π‘Œ is equal to angle 𝐴, angle 𝑍 is equal to angle 𝐷. So therefore, we have the proof angle-angle-angle. The triangles πΆπ‘Œπ‘ and triangle π΄π·π‘Œ 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‘Œ, 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 𝑍𝐢 and 𝑍𝐷 are the same. Well, we also know that the perimeter of triangle 𝐴𝐷𝐢 is 33 centimeters. So therefore, 𝐴𝐢 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 π΄π‘Œ. 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 𝐷𝐢 is equal to two 𝐷𝑍 because we’re told that 𝑍𝐢 and 𝑍𝐷 are in fact the same. Therefore, we know that 𝐴𝐢 is equal to two πΆπ‘Œ. And we can write it as two πΆπ‘Œ is equal to 𝐴𝐢. Therefore, πΆπ‘Œ is gonna be equal to 𝐴𝐢 over two, which makes sense because if we think it’s twice as large, then πΆπ‘Œ is gonna be half the length of it.

So Therefore, πΆπ‘Œ is gonna be equal to 16 over two, which is equal to eight centimeters. Okay, great, so now we’ve got πΆπ‘Œ, let’s find π΄π‘Œ because that’s what we’re looking for. Well, the line segment π΄π‘Œ is going to be to 𝐴𝐢, cause that’s the total line, minus πΆπ‘Œ, 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 𝐴𝐢 was twice πΆπ‘Œ. So we’d expect the line segment π΄π‘Œ to be the same as πΆπ‘Œ so both are eight centimeters.

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