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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