Prove triangle-similarity theorems, including proportional segments and the Pythagorean Theorem via similarity.
Derive the Pythagorean Theorem from altitude similarity
Problem
In right triangle \(ABC\), altitude \(CD\) divides hypotenuse \(AB~=~25\) into \(AD~=~9\) and \(DB~=~16\). Use each small triangle's similarity to the original to derive \(AC^{2}~=~AB\cdot~AD\) and \(BC^{2}~=~AB\cdot~DB\), evaluate \(\text{both}\), and add them to recover \(AB^{2}\).
Altitude similarity turns each leg into the geometric mean of the hypotenuse and that leg’s adjacent projection. We’ll derive one leg-square relation from each smaller triangle, keep the projection pairings straight, evaluate both squares, and add the relations so the Pythagorean identity emerges.
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