In the figure above, points $P$ and $Q$ lie on a circle with center $O$. If triangle $OPQ$ is a right triangle with right angle at $P$, and $OP=5$, $OQ=13$, what is the value of $PQ$ (denoted as $s$)?

• A. $8$
• B. $10$
• C. $18$
• D. $12$

Answer: (D)

Verified step by step guidance

1. Identify the given elements: triangle OPQ is a right triangle with the right angle at P, and the lengths OP = 5 and OQ = 13 are given.
2. Recall the Pythagorean theorem for a right triangle, which states that the square of the hypotenuse equals the sum of the squares of the other two sides. Since the right angle is at P, the side opposite P (OQ) is the hypotenuse.
3. Set up the Pythagorean theorem equation: \(OQ^2 = OP^2 + PQ^2\), where \(PQ\) is the side we want to find (denoted as \(s\)).
4. Substitute the known values into the equation: \$13^2 = 5^2 + s^2$.
5. Solve for \(s^2\) by isolating it: \(s^2 = 13^2 - 5^2\). Then, take the square root of both sides to find \(s\).