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Rule

Geometric Mean Altitude Theorem

Given a right triangle, if an altitude is drawn from the vertex of the right angle to the hypotenuse, then the measure of this altitude is the geometric mean between the measures of the two segments formed on the hypotenuse.

Based on the diagram above and by definition of the geometric mean, the following relation holds true.

$C D^{2} = A D \cdot B D$ or $\frac{C D}{A D} = \frac{B D}{C D}$

The Geometric Mean Altitude Theorem is also known as the Right Triangle Altitude Theorem and the Geometric Mean Theorem.

Proof

According to the Right Triangle Similarity Theorem, the two triangles formed by the altitude

\overline{C D}

are similar.

△ C B D \sim △ A C D

Then, by definition of similar triangles, the lengths of corresponding sides are proportional.

$\frac{C D}{A D} = \frac{B D}{C D}$

Applying the Properties of Equality, this proportion can be rewritten without fractions.

$C D^{2} = A D \cdot B D$

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