In addition to the length of the altitude being the geometric mean of the lengths of the two hypotenuse segments, there are additional ratios created by the three similar right triangles. In this section, you will investigate relationships between the lengths of the legs and the hypotenuse segments.

For this, we will need to know a few more vocabulary terms:

In the figure above the altitude creates two similar triangles, the red triangle, ∆ABD, is similar to the blue triangle, ∆BCD.  Since they are similar they have similar ratios.

Investigating Ratios of Legs and Hypotenuse Segments

1. After clicking on the link below, scroll down below the video to “So what does this have to do with right similar triangles?”
2. Read the information and then click on “Little”, then “Middle”, and “Largest” to rotate the triangles. 

Right Similar Triangles Applet

What patterns do you notice?

Answer the following in your notes:

1. In terms of geometric mean, explain what the two equal ratios in the figure indicate about the relationship between the leg, its adjacent hypotenuse segment, and the hypotenuse.
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Check Your Answer
Each leg is the geometric mean of its adjacent hypotenuse segment and the hypotenuse:

Hypotenuse Segment Leg = Leg Hypotenuse
2. Use what you know about similar triangles to explain why these ratios are equal. Illustrate your statement with a picture (see Section 1 if you need assistance).
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Check Your Answer
When each of the smaller right triangles created by the altitude is compared to the original right triangle, the hypotenuse segment (leg length on the smaller right triangle) is a corresponding part to the leg of the original right triangle.

The leg of the original triangle becomes the hypotenuse of the smaller right triangles. Each of these sides of the smaller right triangles is a corresponding part to the hypotenuse of the original right triangle.