Pearson Geometry Common Core, 2011
PG
Pearson Geometry Common Core, 2011 View details
4. Similarity in Right Triangles
Continue to next subchapter

Exercise 43 Page 466

Look for similar triangles.

See solution.

Practice makes perfect

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.

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–ł ABC~ â–ł ACD â–ł ABC~ â–ł CBD

The 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. △ 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.

Proof of the Second Part

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.

Completed Proof

2 &Given:&& â–ł ABC is a right triangle with & && altitude to the hypotenuseCD &Prove:&& AB/AC=AC/ADandAB/BC=BC/DB Proof: