Final answer:
To find the length PZ, set up the proportion using the similarity of the triangles and solve for x. Disregarding the negative value, x = 3, and substituting back into PZ = x + 7 gives PZ = 10 units.
Step-by-step explanation:
If AQPZ is similar to ASRZ (denoted by AQPZ ~ ASRZ), we can set up proportions to find the length of PZ. Since triangle similarity means that corresponding sides are in proportion, we have:
· QZ/SZ = PZ/RZ
· 15/(x + 3) = (x + 7)/4
Now, we can cross-multiply to solve for x:
· 4 × 15 = (x + 3)(x + 7)
· 60 = x² + 10x + 21
Next, we solve the quadratic equation for x:
· x² + 10x + 21 - 60 = 0
· x² + 10x - 39 = 0
This equation factors to:
· (x + 13)(x - 3) = 0
Therefore, we get two potential solutions for x, x = -13 or x = 3. Since we are dealing with a length, we can disregard the negative value, leaving x = 3.
Now, we can substitute x back into PZ = x + 7:
· PZ = 3 + 7
· PZ = 10
The requested length PZ is 10 units.