geometry (please explain)

Question

Roman is using the figure shown below to prove Pythagorean Theorem using triangle similarity.

In the given triangle ABC, angle A is 90° and segment AD is perpendicular to segment BC.

Part A: Identify a pair of similar triangles.

Part B: Explain how you know the triangles from Part A are similar.

Part C: If DB = 9 and DC = 4, find the length of segment DA. Show your work.

Answer 1

Answer 2

Angle C is common to both the large composite right triangle and to the smaller right triangle on the right side

both those two triangles also share 90 degrees as another angle, angle A for the composite largest triangle and angle D for the smallest triangle. if they share 2 angles, they also share the 3rd angle. They're two similar triangles. Triangles ABC and ADC. Angle ADC = Angle ABC, Angle BAC = Angle ADC, and Angle ACD = Angle ACB,

Triangle BDA is also a right triangle, similar to both the other right triangles, by the same reasoning.

IF DB =9 and DC = 4, then BC = 9+4=13

rotate the 2 smaller triangles so they have bases 4 and AD

with heights AD and 9

then the ratios are the same: base/height = AD/9= 4/AD

cross multiply

AD^2 = 9(4) =36

AD = sqr36

= 6

there are 100+ proofs of the Pythagorean Theorem. One semi-famous one was by former President James A. Garfield, a similar geometric proof. Garfield used to be in the House of Representative when he came up with it. Google proofs of Pythagorean Theorem and see endless proofs