Video: Finding the Length of a Line Segment Using Similarity of Triangles

Given that ๐ท๐ธ = 74 m, ๐ธ๐ต = 32 m, and ๐ธ๐ด = 48 m, find the length of ๐ถ๐ด.

03:49

Video Transcript

Given that ๐ท๐ธ is 74 meters, ๐ธ๐ต is 32 meters, and ๐ธ๐ด is 48 meters, find the length of ๐ถ๐ด.

First, letโ€™s add the information weโ€™ve been given in the question onto the diagram. So we have two right-angled triangles in which weโ€™ve been given various lengths. And weโ€™re asked to find the length of ๐ถ๐ด which is, in fact, the length of the two hypotenuses of the triangle joined together, ๐ถ๐ธ and ๐ธ๐ด.

So in order to answer the question, weโ€™re going to need to work out the length of ๐ถ๐ธ first of all. So letโ€™s think about how weโ€™re going to do this. Weโ€™ve been given two triangles, both right-angled triangles. And as I said, weโ€™ve been given lengths in each of them.

This suggests that we might want to approach this problem using similar triangles. We canโ€™t just assume that the two triangles are similar. We need to show it first. In order for two triangles to be similar, they need to have all three angles the same. Both triangles have a right angle. So first of all we can say that angle ๐ถ๐ท๐ธ is congruent to angle ๐ด๐ต๐ธ.

Letโ€™s look at the pair of angles marked in green. This pair of angles are vertically opposite one another, as theyโ€™re formed by a pair of intersecting straight lines. Therefore, these two angles are equal. So we have that angle ๐ท๐ธ๐ถ is congruent to angle ๐ต๐ธ๐ด. Now it follows automatically that the third angle in these two triangles is also equal because the angle sum in a triangle is always 180 degrees.

And if the other two angles in each triangle are equal, then the third must also be the same. For this reason, itโ€™s actually sufficient to show that two of the angles in a pair of triangles are the same in order to conclude that the two triangles are similar. So we can conclude that triangle ๐ถ๐ท๐ธ is similar to triangle ๐ด๐ต๐ธ.

Now how does this help us with answering the question? Well, remember that if two triangles are similar, then they have proportional side lengths. This means that if I divide a side in the larger triangle by a corresponding side in the smaller triangle, then I always get the same ratio. Letโ€™s substitute in the values for the sides we know.

We have 74 divided by 32 for ๐ท๐ธ and ๐ต๐ธ, the horizontal side of the two triangles. And then we have ๐ถ๐ธ divided by 48 for the two hypotenuses. This gives an equation that we can solve in order to find ๐ถ๐ธ. We need to multiply both sides of the equation by 48. So ๐ถ๐ธ is 48 multiplied by 74 over 32, which is 111.

Now we havenโ€™t finished the question yet because, remember, we werenโ€™t asked just to calculate ๐ถ๐ธ, we were asked to calculate ๐ถ๐ด, which is this full length here, the sum of ๐ถ๐ธ and ๐ธ๐ด. But we know both of these lengths. So we just need to add them together.

We have that ๐ถ๐ด is equal to 111 plus 48, which gives us our answer of 159 meters. Remember, we used similar triangles in this question. But we couldnโ€™t just assume the two triangles were similar. We had to prove it first by giving the reasoning for why their angles were equal.

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