+45 Find RZ in the figure at right, if PR = 9, QZ = 4, PQ = 6, and PZ bisects ∠QPR.
+1565 Since PZ bisects ∠QPR, ∠QPZ=∠RPZ. Also, since QZ=4 and PZ=2RP=29, we can find the length of PZ using the Pythagorean Theorem:
QZ2+PZ2=PR2.
Substituting the given values, we have
42+(29)2=92.
Solving for PZ, we get PZ=215.
Since ∠QPR is a right angle and ∠RPZ=∠QPZ, ∠RPZ and ∠QPZ are each complementary to ∠QPR. Therefore,
∠QPZ+∠RPZ=90∘.
Substituting the given values, we have
∠QPZ+∠RPZ=90∘,
so ∠QPZ=∠RPZ=290∘=45∘.
Now, since ∠RPZ=∠QPZ=45∘, triangle RPZ is a 45-45-90 triangle, and RZ=15/sqrt(2).
