Difference between revisions of "1953 AHSME Problems/Problem 14"

(Problem 14)
(Problem 14)
Line 7: Line 7:
 
\textbf{(C)}\ p+q\text{ can be less than }\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}  </math>
 
\textbf{(D)}\ p-q\text{ can be less than }\overline{PQ}\ \textbf{(E)}\ \text{none of these}  </math>
 +
 +
==Solution==
 +
 +
We will test each option to see if it can be true or not.
 +
<cmath>\textbf{(A)}\ p-q\text{ can be equal to }\overline{PQ}</cmath>
 +
If circle <math>Q</math> is inside circle <math>P</math> and it is tangent to circle <math>P</math>, then <math>PQ</math> is <math>p-q</math>.
 +
<cmath>\textbf{(B)}\ p+q\text{ can be equal to }\overline{PQ}</cmath>
 +
If circle <math>Q</math> is outside circle <math>P</math> and it is tangent to circle <math>P</math>, then <math>PQ</math> is <math>p+q</math>.
 +
<cmath>\textbf{(C)}\ p+q\text{ can be less than }\overline{PQ}</cmath>
 +
If circle <math>Q</math> is outside circle <math>P</math> and it is not tangent to circle <math>P</math>, then <math>PQ</math> is greater than <math>p+q</math>.
 +
<cmath>\textbf{(D)}\ p-q\text{ can be less than }\overline{PQ}</cmath>
 +
If circle <math>Q</math> is inside circle <math>P</math> and it is not tangent to circle <math>P</math>, then <math>PQ</math> is greater than <math>p-q</math>.
 +
Since options A, B, C, and D can be true, the answer must be <math>\boxed{E}</math>.
  
 
==See Also==
 
==See Also==

Revision as of 14:43, 15 July 2018

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 $p-q$. \[\textbf{(B)}\ p+q\text{ can be equal to }\overline{PQ}\] 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 $\boxed{E}$.

See Also

1953 AHSC (ProblemsAnswer KeyResources)
Preceded by
Problem 12
Followed by
Problem 14
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50
All AHSME Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png