Question

What is the measure of n?

n
20
m
5
n =
= [?]
Give your answer in simplest form.

Answer 1

n = 10

Explanation:

The right triangles in this figure are all similar, so the ratios of corresponding sides are proportional. Here, we're interested in the ratio of the short side to the long side of the smaller two triangles. That gives rise to a geometric mean rule telling you that n is the geometric mean of the segments of the long hypotenuse.

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The applicable geometric mean rule for n is ...

n/5 = 20/n

n = √(5×20)

n = √100 = 10

The measure of n is 10.

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Additional comment

There are two other applicable geometric mean rules. The one that applies to m is that ratios of hypotenuse to short side are proportional

m/5 = (5+20)/m

m = √(5(25)) = 5√5