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Geometry and Trigonometry Difficulty: Hard

Points $Q$ and $R$ lie on a circle with center $P$ . The radius of this circle is $9$ inches. Triangle $P Q R$ has a perimeter of $31$ inches. What is the length, in inches, of $\bar{Q R}$ ?

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Explanation

Choice B is correct. Since it's given that $P$ is the center of a circle with a radius of $9$ inches, and that points $Q$ and $R$ lie on that circle, it follows that $\bar{P Q}$ and $\bar{R P}$ of triangle $P Q R$ each have a length of $9$ inches. Let the length of $\bar{Q R}$ be $x$ inches. It follows that the perimeter of triangle $P Q R$ is $9 + 9 + x$ inches. Since it's given that the perimeter of triangle $P Q R$ is $31$ inches, it follows that $9 + 9 + x = 31$ , or $18 + x = 31$ . Subtracting $18$ from both sides of this equation gives $x equals 13$ . Therefore, the length, in inches, of $\bar{Q R}$ is $13$ .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.