Find RZ in the figure at right, if PR = 9, QZ = 4, PQ = 6, and PZ bisects ∠QPR.

LuckyLemonie Nov 18, 2023

#1**0 **

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).

bader Nov 18, 2023