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Look for similar triangles.
See solution.
According to Theorem 7-3, the altitude to the hypotenuse cuts a right triangle into two triangles that are similar to each other and to the original triangle.
The corollary's claim can be proven using that by definition corresponding sides of similar triangles are proportional.
The first similarity gives the first part of the claim. △ ABC~ △ ACD ⇓ AB/AC=AC/AD This proportion means that the length of leg AC of the triangle is the geometric mean of the length of the hypotenuse AB and the length of the adjacent segment of the hypotenuse AD.
The second similarity gives the second part of the claim. △ ABC~ △ CBD ⇓ AB/BC=BC/DB This proportion means that the length of leg BC of the triangle is the geometric mean of the length of the hypotenuse AB and the length of the adjacent segment of the hypotenuse DB. We can summarize the steps above in a flow proof.
2 &Given:&& â–ł ABC is a right triangle with & && altitude to the hypotenuseCD &Prove:&& AB/AC=AC/ADandAB/BC=BC/DB Proof: