The angle bisectors of triangle XYZ meet at point P. What is PJ?

Explanation:

Segment PJ is perpendicular to segment YZ

So triangle PJZ is a right triangle with the known leg ZJ = 12, unknown leg PJ = x, and known hypotenuse PZ = 20.

Let's say a = 12, b = x, c = 20.

Use the Pythagorean theorem to find x

a^2 + b^2 = c^2

12^2 + x^2 = 20^2

144 + x^2 = 400

x^2 = 400-144

x^2 = 256

x = sqrt(256)

x = 16

Therefore segment PJ is 16 units long.

Side note: the angle bisectors won't always form 90 degree angles with the opposite sides. These angle bisectors are also altitude lines because of the 90 degree angles that form.