Question

Find the value of y.

Answer 1

y = √55

Explanation:

All the triangles are similar, so the ratio of short side to hypotenuse is the same for all:

5/y = y/(5+6)

y^2 = 55

y = √55

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Comment on the geometry

There are three "geometric mean" relationships that apply to this geometry.

1. the altitude BD is the geometric mean of BC and BA (√30)
2. the long side of the large triangle is the geometric mean of the longer segment of the hypotenuse and the whole hypotenuse (x = √66)
3. the short side of the large triangle is the geometric mean of the shorter segment of the hypotenuse and the whole hypotenuse (y = √55)

If you were aware of the last of these relationships, you could write down the answer without any "work."

y = √(5(5 +6)) = √55

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The geometric mean is the n-th root of the product of n numbers. When there are 2 numbers, it is the square root of their product.