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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 CD is drawn from the vertex of the right triangle at C to the hypotenuse at D, the following relations hold true.

AC/AD = AB/AC [0.4cm] CB/DB = AB/CB or AC^2 = AD * AB CB^2 = DB * AB

Proof

According to the Right Triangle Similarity Theorem, the two triangles formed by the altitude CD are similar to △ ABC. △ ABC ~ △ ACD △ ABC ~ △ CBD This means that the corresponding parts of the triangles can be identified.

△ ABC ~ △ ACD	△ ABC ~ △ CBD
AB and AC AC and AD CB and CD	AB and CB AC and CD CB and DB

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

△ ABC ~ △ ACD	△ ABC ~ △ CBD
AC/AD = AB/AC	CB/DB = AB/CB

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.

AC^2 = AD * AB CB^2 = DB * AB

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