Right Triangle Proportions — Practice Geometry Questions
When you draw an altitude to the hypotenuse of a right triangle, you create two new triangles with some interesting properties: first, they are also right triangles, and second, they are similar to each other and to the original right triangle.
The following practice questions ask you to use ‘mean proportionals’ to get to the solutions—no mean feat!
Practice questions
Use the figure and given information to solve the following problems.

If GI = 9 and RI = 4, find the length of

If RT = 8 and GI = 12, find the length of
Answers and explanations

6
When you draw an altitude to the hypotenuse of a right triangle, you create two right triangles that are similar to each other and to the original right triangle. Because these triangles are similar, you can set up proportions relating the corresponding sides.
The altitude to the hypotenuse of a right triangle is the mean proportional between the two segments that the hypotenuse is divided into:
In the figure, this would mean that
Crossmultiplying gives you the following:

4
The leg of a right triangle is the mean proportional between the hypotenuse and the projection of the leg on the hypotenuse:
In the figure, this would mean that
Crossmultiplying gives you the following: