+0  
 
+1
21
1
avatar+45 

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

 Nov 18, 2023
 #1
avatar+1612 
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).

 Nov 18, 2023

4 Online Users

avatar