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Difference between revisions of "1991 AHSME Problems/Problem 22"

(Created page with "== Problem == <asy> draw(circle((0,0),10),black+linewidth(.75)); MP(")",(0,0),S); </asy> Two circles are externally tangent. Lines <math>\overline{PAB}</math> and <math>\overli...")
 
m (→‎Problem)
Line 1: Line 1:
 
== Problem ==
 
== Problem ==
 
<asy>
 
<asy>
−
draw(circle((0,0),10),black+linewidth(.75));
+
draw(circle((0,6sqrt(2)),2sqrt(2)),black+linewidth(.75));
−
MP(")",(0,0),S);
+
draw(circle((0,3sqrt(2)),sqrt(2)),black+linewidth(.75));
 +
draw((-8/3,16sqrt(2)/3)--(-4/3,8sqrt(2)/3)--(0,0)--(4/3,8sqrt(2)/3)--(8/3,16sqrt(2)/3),dot);
 +
MP("B",(-8/3,16*sqrt(2)/3),W);MP("B'",(8/3,16*sqrt(2)/3),E);
 +
MP("A",(-4/3,8*sqrt(2)/3),W);MP("A'",(4/3,8*sqrt(2)/3),E);
 +
MP("P",(0,0),S);
 
</asy>
 
</asy>
  
  
−
Two circles are externally tangent. Lines <math>\overline{PAB}</math> and <math>\overline{PA'B'}</math> are common tangents with <math>A</math> and <math>A'</math> on the smaller circle <math>B</math> and <math>B'</math> on the larger circle. If <math>PA=AB=4</math>, then the are of the smaller circle is
+
Two circles are externally tangent. Lines <math>\overline{PAB}</math> and <math>\overline{PA'B'}</math> are common tangents with <math>A</math> and <math>A'</math> on the smaller circle <math>B</math> and <math>B'</math> on the larger circle. If <math>PA=AB=4</math>, then the area of the smaller circle is
  
 
<math>\text{(A) } 1.44\pi\quad
 
<math>\text{(A) } 1.44\pi\quad

Revision as of 14:48, 28 September 2014

Problem

[asy] draw(circle((0,6sqrt(2)),2sqrt(2)),black+linewidth(.75)); draw(circle((0,3sqrt(2)),sqrt(2)),black+linewidth(.75)); draw((-8/3,16sqrt(2)/3)--(-4/3,8sqrt(2)/3)--(0,0)--(4/3,8sqrt(2)/3)--(8/3,16sqrt(2)/3),dot); MP("B",(-8/3,16*sqrt(2)/3),W);MP("B'",(8/3,16*sqrt(2)/3),E); MP("A",(-4/3,8*sqrt(2)/3),W);MP("A'",(4/3,8*sqrt(2)/3),E); MP("P",(0,0),S); [/asy]


Two circles are externally tangent. Lines $\overline{PAB}$ and $\overline{PA'B'}$ are common tangents with $A$ and $A'$ on the smaller circle $B$ and $B'$ on the larger circle. If $PA=AB=4$, then the area of the smaller circle is

$\text{(A) } 1.44\pi\quad \text{(B) } 2\pi\quad \text{(C) } 2.56\pi\quad \text{(D) } \sqrt{8}\pi\quad \text{(E) } 4\pi$

Solution

$\fbox{B}$

See also

1991 AHSME (Problems • Answer Key • Resources)
Preceded by
Problem 21 		Followed by
Problem 23
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
All AHSME Problems and Solutions

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