Difference between revisions of "2000 AIME II Problems/Problem 10"
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== Problem == | == Problem == | ||
| − | A [[circle]] is [[inscribe]]d in [[quadrilateral]] <math>ABCD</math>, [[tangent]] to <math>\overline{AB}</math> at <math>P</math> and to <math>\overline{CD}</math> at <math>Q</math>. Given that <math>AP=19</math>, <math>PB=26</math>, <math>CQ=37</math>, and <math>QD=23</math>, find the [[square]] of the [[radius]] of the circle. | + | A [[circle]] is [[inscribe]]d in [[quadrilateral]] <math>ABCD</math>, [[tangent]] to <math>\overline{AB}</math> at <math>P</math> and to <math>\overline{CD}</math> at <math>Q</math>. Given that <math>AP=19</math>, <math>PB=26</math>, <math>CQ=37</math>, and <math>QD=23</math>, find the [[Perfect square|square]] of the [[radius]] of the circle. |
== Solution == | == Solution == | ||
Revision as of 08:10, 20 March 2008
Problem
A circle is inscribed in quadrilateral 
, tangent to 
at 
and to 
at 
. Given that 
, 
, 
, and 
, find the square of the radius of the circle.
Solution
Call the center of the circle 
. By drawing the lines from tangent to the sides and from to the vertices of the quadrilateral, eight congruent right triangles are formed.
Thus, 
, or 
.
Take the 
of both sides and use the identity for 
to get 
.
Use the identity for again to get 
.
Solving gives 
.
| 2000 AIME II (Problems • Answer Key • Resources) | ||
| Preceded by Problem 9 |
Followed by Problem 11 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
