How to Calculate the Geometric Mean in Triangles

The geometric mean of two numbers is the square root of their product. In a right triangle, the altitude to the hypotenuse is the geometric mean of the two segments it creates, which is where the right triangle similarity relationships come from.

Worked examples: geometric mean in a right triangle

Practice questions on the geometric mean in triangles

1. If an altitude to the hypotenuse of a right triangle divides the hypotenuse into segments of (5 text{ cm}) and (20 text{ cm}), what is the length of the altitude?
2. In a right triangle with a hypotenuse of (13 text{ cm}) and one segment of (5 text{ cm}), find the length of the leg adjacent to the (5 text{ cm}) segment.

1. ( h = sqrt{5 times 20} = sqrt{100} = 10 text{ cm}).
2. Leg length (= sqrt{13 times 5} approx 8.06 text{ cm}).

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