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Rule

Geometric Mean Leg Theorem

In a right triangle, if the altitude is drawn from the vertex of the right angle to the hypotenuse, then the measure of each leg of the triangle is the geometric mean between the length of the hypotenuse and the length of the segment formed on the hypotenuse adjacent to the leg.

Based on the given triangle, where the altitude $\overline{C D}$ is drawn from the vertex of the right triangle at $C$ to the hypotenuse at $D,$ the following relations hold true.

⎩ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎧ \frac{A C}{A D} = \frac{A B}{A C} \frac{C B}{D B} = \frac{A B}{C B} or {A C^{2} = A D \cdot A B C B^{2} = D B \cdot A B

Proof

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

\overline{C D}

are similar to

△ A B C .

△ A B C \sim △ A C D △ A B C \sim △ C B D

This means that the corresponding parts of the triangles can be identified.

$△ A B C \sim △ A C D$	$△ A B C \sim △ C B D$
$\overline{A B} and \overline{A C}$ $\overline{A C} and \overline{A D}$ $\overline{C B} and \overline{C D}$	$\overline{A B} and \overline{C B}$ $\overline{A C} and \overline{C D}$ $\overline{C B} and \overline{D B}$

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

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

Note that for any two pairs of corresponding sides a similar proportion can be obtained. Now, applying the Properties of Equality, the proportion can be rewritten without fractions.

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

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