SOLUTION: What is the proof for Theorem 7.6 Geometric Mean (Altitude): In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments

Question 407117: What is the proof for Theorem 7.6 Geometric Mean (Altitude): In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments
Answer by richard1234(7193) (Show Source): You can put this solution on YOUR website!
Yes, the altitude divides the hypotenuse into two segments:

I think you want to show that the geometric mean of the two segments equals the length of the altitude, which is a famous theorem.

The proof comes from the fact that triangles ABD and CBD are similar, which I've already shown. Therefore, we can let , , and . The geometric mean of AB and DC is given by

which is the same as BD, so we're done.
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