Plese help to find the soloutions In the diagram ABC 90 deg

Plese help to find the soloutions

In the diagram, ABC = 90 degree, AB = 16 cm, BC = 12 cm, BP =x cm and the point Q on the line C is the image point of B under the reflection in the line AP. Calculate, without the use of a calculator, the exact value of x, tan CAP.

Solution

we apply pythagorean formula on the triangle ABC

AB^2 + BC^2 = AC^2

plugging the values

16^2 + 12^2 = AC^2

AC = 20

applying cosine rule to find angle C

16^2 =12^2 + 20^2 - 2*12*20 cos C

C = 53.13 degrees

angle A = 180 - ( 53.13 + 90) = 36.87

therefore, tan CAP = tan 18.43 = 1/3

tan 71.56 = 16/x

x = 3