Difference between revisions of "2006 AMC 10B Problems/Problem 24"
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== Problem == | == Problem == | ||
[[Circle]]s with centers <math>O</math> and <math>P</math> have radii <math>2</math> and <math>4</math>, respectively, and are externally tangent. Points <math>A</math> and <math>B</math> on the circle with center <math>O</math> and points <math>C</math> and <math>D</math> on the circle with center <math>P</math> are such that <math>AD</math> and <math>BC</math> are common external tangents to the circles. What is the area of the [[concave]] [[hexagon]] <math>AOBCPD</math>? | [[Circle]]s with centers <math>O</math> and <math>P</math> have radii <math>2</math> and <math>4</math>, respectively, and are externally tangent. Points <math>A</math> and <math>B</math> on the circle with center <math>O</math> and points <math>C</math> and <math>D</math> on the circle with center <math>P</math> are such that <math>AD</math> and <math>BC</math> are common external tangents to the circles. What is the area of the [[concave]] [[hexagon]] <math>AOBCPD</math>? | ||
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<asy>size(200);defaultpen(linewidth(0.8)); | <asy>size(200);defaultpen(linewidth(0.8)); | ||
pair X=(-6,0), O=origin, P=(6,0), B=tangent(X, O, 2, 1), A=tangent(X, O, 2, 2), C=tangent(X, P, 4, 1), D=tangent(X, P, 4, 2); | pair X=(-6,0), O=origin, P=(6,0), B=tangent(X, O, 2, 1), A=tangent(X, O, 2, 2), C=tangent(X, P, 4, 1), D=tangent(X, P, 4, 2); |
Revision as of 17:45, 19 April 2021
Problem
Circles with centers and have radii and , respectively, and are externally tangent. Points and on the circle with center and points and on the circle with center are such that and are common external tangents to the circles. What is the area of the concave hexagon ?
Solution
File is too big, so go to https://www.imgur.com/a/7aphGaa
Sorry for the wrong point names, I didn't know how to change them.
Since a tangent line is perpendicular to the radius containing the point of tangency, .
Construct a perpendicular to that goes through point . Label the point of intersection .
Clearly is a rectangle, so and . By the Pythagorean Theorem, .
The area of is . The area of is , so the area of quadrilateral is . Using similar steps, the area of quadrilateral is also . Therefore, the area of hexagon is .
See also
2006 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 23 |
Followed by Problem 25 | |
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 | ||
All AMC 10 Problems and Solutions |
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