1953 AHSME Problems/Problem 14

Problem 14

Given the larger of two circles with center $P$ and radius $p$ and the smaller with center $Q$ and radius $q$ . Draw $PQ$ . Which of the following statements is false?

$\textbf{(A)}\ p-q\text{ can be equal to }\overline{PQ}\\ \textbf{(B)}\ p+q\text{ can be equal to }\overline{PQ}\\ \textbf{(C)}\ p+q\text{ can be less than }\overline{PQ}\\ \textbf{(D)}\ p-q\text{ can be less than }\overline{PQ}\\ \textbf{(E)}\ \text{none of these}$

Solution

We will test each option to see if it can be true or not. $\textbf{(A)}\ p-q\text{ can be equal to }\overline{PQ}$ If circle $Q$ is inside circle $P$ and it is tangent to circle $P$ , then $PQ$ is $p-q$ . [asy] pair P = (0,0); pair Q = (3,0); draw(Circle(P, 4)); draw(Circle(Q, 1)); draw(P -- (4,0)); dot(P); dot(Q); [/asy] $\textbf{(B)}\ p+q\text{ can be equal to }\overline{PQ}$ If circle $Q$ is outside circle $P$ and it is tangent to circle $P$ , then $PQ$ is $p+q$ . $\textbf{(C)}\ p+q\text{ can be less than }\overline{PQ}$ If circle $Q$ is outside circle $P$ and it is not tangent to circle $P$ , then $PQ$ is greater than $p+q$ . $\textbf{(D)}\ p-q\text{ can be less than }\overline{PQ}$ If circle $Q$ is inside circle $P$ and it is not tangent to circle $P$ , then $PQ$ is greater than $p-q$ . Since options A, B, C, and D can be true, the answer must be $\boxed{E}$ .

1953 AHSC (Problems • Answer Key • Resources)
Preceded by Problem 12	Followed by Problem 14
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1953 AHSME Problems/Problem 14

Problem 14

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