Question

In a triangle PQR, $\angle PRQ = 90^\circ$. What is $PR + RQ$?
Statement A
A. The diameter of the incircle is $10$.

Statement B
B. The diameter of the circumcircle is $18$.

• A. The question can be answered by one of the statements alone but not by the other.
• B. The question can be answered by using either statement alone.
• C. The question can be answered by using both the statements together, but cannot be answered by using either statement alone.
• D. The question cannot be answered even by using both statements together.

Hint:

In right triangles, inradius and circumradius formulas combined can yield sums of legs if hypotenuse is known.

Answer 1

The Correct Option is C

From Statement A: Inradius $r = 5$, but without another dimension, $PR+RQ$ cannot be determined.
From Statement B: Circumradius $R = 9$ in a right triangle means hypotenuse $PQ = 18$, but $PR+RQ$ still cannot be found directly.
Combining both: In a right triangle, $r = \frac{PR + RQ - PQ}{2}$. Knowing $r = 5$ and $PQ = 18$, we solve for $PR + RQ = 28$. Thus, both statements together are needed.