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.