Point P is the center of the circle in the figure above. If triangle $APB$ is a right triangle with right angle at $A$, and $AP=5$, $PB=13$, what is the value of $x$ if $AB=x$?

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

Answer: (D)

Verified step by step guidance

1. Identify the given elements: triangle \( A P B \) is a right triangle with the right angle at \( A \), and the lengths \( A P = 5 \) and \( P B = 13 \). We need to find \( A B = x \).
2. Recall the Pythagorean theorem for a right triangle: the square of the hypotenuse equals the sum of the squares of the other two sides. Since the right angle is at \( A \), the side opposite \( A \) (which is \( P B \)) is the hypotenuse.
3. Set up the Pythagorean theorem equation: \( (A P)^2 + (A B)^2 = (P B)^2 \). Substitute the known values: \( 5^2 + x^2 = 13^2 \).
4. Simplify the equation: \( 25 + x^2 = 169 \).
5. Isolate \( x^2 \) and solve for \( x \): \( x^2 = 169 - 25 \), then \( x = \sqrt{144} \).