Question

Seth 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.

The figure shows triangle ABC with right angle at A and segment AD. Point D is on side BC.

Part A: Identify a pair of similar triangles. (2 points)

Part B: Explain how you know the triangles from Part A are similar. (4 points)

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

Answer 1

Part A: Triangles ADB and ADC are similar.

Part B:
- Angle A is common to both triangles.
- Angle ADB and ADC are both right angles.
- Therefore, by AA similarity, triangles ADB and ADC are similar.

Part C:
Using the Pythagorean Theorem in triangle ADB:
(AD)^2 + (DB)^2 = (AB)^2
(AD)^2 + 9^2 = AC^2
(AD)^2 + 81 = AC^2

Using the Pythagorean Theorem in triangle ADC:
(AD)^2 + (DC)^2 = (AC)^2
(AD)^2 + 4^2 = AC^2
(AD)^2 + 16 = AC^2

Setting the two equations equal to each other:
(AD)^2 + 81 = (AD)^2 + 16
65 = 16
(AD)^2 = 65
AD = √65

Therefore, the length of segment DA is √65.