1953 AHSME Problems/Problem 14
Problem 14
Given the larger of two circles with center 
and radius 
and the smaller with center 
and radius 
. Draw 
. Which of the following statements is false?
Solution
We will test each option to see if it can be true or not. Links to diagrams are provided.
![\[\textbf{(A)}\ p-q\text{ can be equal to }\overline{PQ}\]](http://latex.artofproblemsolving.com/1/e/6/1e688d43d1c33da119b8eea5612c11048fa134f9.png)
Let circle be inside circle and tangent to circle , and the point of tangency be 
. 
, and 
, so 
Diagram A
![\[\textbf{(B)}\ p+q\text{ can be equal to }\overline{PQ}\]](http://latex.artofproblemsolving.com/9/a/9/9a95d9d55e67d554764453c58f9aff43f1f4fd5f.png)
Let circle be outside circle and tangent to circle , and the point of tangency be . , and , so 
Diagram B
![\[\textbf{(C)}\ p+q\text{ can be less than }\overline{PQ}\]](http://latex.artofproblemsolving.com/f/e/2/fe2af986dceb9f4a694da53d16e915bcf6a049db.png)
Let circle be outside circle and not tangent to circle , and the intersection of 
with the circles be and 
respectively. and 
, and 
, so 
Diagram C
![\[\textbf{(D)}\ p-q\text{ can be less than }\overline{PQ}\]](http://latex.artofproblemsolving.com/6/9/b/69b1f337d24bebff831aa44b88406c83002a5319.png)
Let circle be inside circle and not tangent to circle , and the intersection of with the circles be and respectively. and , and 
, so 
, and 
, so 
Diagram D
Since options A, B, C, and D can be true, the answer must be 
.
See Also
| 1953 AHSC (Problems • Answer Key • Resources) | ||
| Preceded by Problem 12 |
Followed by Problem 14 | |
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| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
