Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

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 (Problems • Answer Key • Resources)
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

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1953_AHSME_Problems/Problem_14&oldid=96257"