### Video Transcript

Given that ๐ท๐ธ is 74 meters, ๐ธ๐ต is 32 meters, and ๐ธ๐ด is 48 meters, find the length of ๐ถ๐ด.

First, letโs add the information weโve been given in the question onto the diagram. So we have two right-angled triangles in which weโve been given various lengths. And weโre asked to find the length of ๐ถ๐ด which is, in fact, the length of the two hypotenuses of the triangle joined together, ๐ถ๐ธ and ๐ธ๐ด.

So in order to answer the question, weโre going to need to work out the length of ๐ถ๐ธ first of all. So letโs think about how weโre going to do this. Weโve been given two triangles, both right-angled triangles. And as I said, weโve been given lengths in each of them.

This suggests that we might want to approach this problem using similar triangles. We canโt 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 ๐ถ๐ท๐ธ is congruent to angle ๐ด๐ต๐ธ.

Letโs look at the pair of angles marked in green. This pair of angles are vertically opposite one another, as theyโre formed by a pair of intersecting straight lines. Therefore, these two angles are equal. So we have that angle ๐ท๐ธ๐ถ is congruent to angle ๐ต๐ธ๐ด. 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, itโs 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 ๐ถ๐ท๐ธ is similar to triangle ๐ด๐ต๐ธ.

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. Letโs substitute in the values for the sides we know.

We have 74 divided by 32 for ๐ท๐ธ and ๐ต๐ธ, the horizontal side of the two triangles. And then we have ๐ถ๐ธ divided by 48 for the two hypotenuses. This gives an equation that we can solve in order to find ๐ถ๐ธ. We need to multiply both sides of the equation by 48. So ๐ถ๐ธ is 48 multiplied by 74 over 32, which is 111.

Now we havenโt finished the question yet because, remember, we werenโt asked just to calculate ๐ถ๐ธ, we were asked to calculate ๐ถ๐ด, which is this full length here, the sum of ๐ถ๐ธ and ๐ธ๐ด. But we know both of these lengths. So we just need to add them together.

We have that ๐ถ๐ด is equal to 111 plus 48, which gives us our answer of 159 meters. Remember, we used similar triangles in this question. But we couldnโt just assume the two triangles were similar. We had to prove it first by giving the reasoning for why their angles were equal.