how wide is the river?

Question

Aldo wants to measure the width of a river. He marks off two right triangles, The base of the larger triangle has a length of 63m

, and the base of the smaller triangle has a length of 32m

. The height of the smaller triangle is 23.6m

. How wide is the river?

Answer 1

There's one key idea that helps us solve this problem: the triangles are similar! How do we know? Because in creating the two triangles, the point where the diagonal line intersects the base of both triangles creates a pair of opposite angles, which have to be the same. They also have a right angle, which is given, because these are both right triangles. And if two of the angles are the same in both triangles, the third one has to be the same (definition of a triangle). Ta-da!

Because the triangles are similar, we know that their sides are proportional. This means that if you multiply the sides of one triangle by a certain number (called a scale factor), we get the sides of the other triangle. To find this number, we have to divide the known sides of both of our triangles.

They give us the base for both, so we can divide 63 by 32, and get... 63/32, or about 1.96875. It's not very exciting, and it doesn't simplify to a nicer fraction, but we'll work with it. The important thing is that 63/32 (or 1.96875) is our scale factor, which means if we multiply that by whatever side in our smaller triangle, we can get the corresponding side length in our bigger triangle. Doing this, we multiply 23.6 by 1.96875, and come out with 46.4625.

Hope this helps!