Right Triangle Proportions — Practice Geometry Questions - dummies

Right Triangle Proportions — Practice Geometry Questions

By Allen Ma, Amber Kuang

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.

image0.png

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

    image1.png

  2. If RT = 8 and GI = 12, find the length of

    image2.png

Answers and explanations

  1. 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:

    image3.png

    In the figure, this would mean that

    image4.png

    Cross-multiplying gives you the following:

    image5.png

  2. 4

    The leg of a right triangle is the mean proportional between the hypotenuse and the projection of the leg on the hypotenuse:

    image6.png

    In the figure, this would mean that

    image7.png

    Cross-multiplying gives you the following:

    image8.png