We are asked to prove Corollary 2 to Theorem 7-3. To do this, let's use that the two smaller triangles are similar to the original triangle.
â–³ ABC~ â–³ ACD
â–³ ABC~ â–³ CBDThe corollary's claim can be proven using that by definition corresponding sides of similar triangles are proportional.
Proof of the First Part
The first similarity gives the first part of the claim.
â–³ ABC~ â–³ 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.
Proof of the Second Part
The second similarity gives the second part of the claim.
â–³ ABC~ â–³ 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.
Completed Proof
2
&Given:&& â–³ ABC is a right triangle with
& && altitude to the hypotenuseCD
&Prove:&& AB/AC=AC/ADandAB/BC=BC/DB
Proof:
