Difference between revisions of "2006 AMC 12B Problems/Problem 15"
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(Another solution , PLEASE edit as I am not sure how to make this clear, and please change to LaTeX. I tried to do it in LaTeX and I'm still new to the software.) |
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<math>2 \cdot \sqrt{32} + \frac{1}{2}\cdot2\cdot \sqrt{32} = 3\sqrt{32} = 12\sqrt{2}</math>, which is half the area of the hexagon, so the area of the entire hexagon is <math>2\cdot12\sqrt{2} = \boxed{(B)} \qquad24\sqrt{2}</math> | <math>2 \cdot \sqrt{32} + \frac{1}{2}\cdot2\cdot \sqrt{32} = 3\sqrt{32} = 12\sqrt{2}</math>, which is half the area of the hexagon, so the area of the entire hexagon is <math>2\cdot12\sqrt{2} = \boxed{(B)} \qquad24\sqrt{2}</math> | ||
+ | |||
+ | == Solution 2 == | ||
+ | |||
+ | ADOP and OPBC are congruent right trapezoids with legs 2 and 4 and with OP equal to 6. Draw an altitude from O to either DP or CP, creating a rectangle with width 2 and base x, and a right triangle with one leg 2, the hypotenuse 6, and the other x. Using | ||
+ | the Pythagorean theorem, x is equal to <math>4\sqrt{2}</math>, and x is also equal to the height of the trapezoid. The area of the trapezoid is thus 1/2(4+2)*4sqrt(2) = 12sqrt(2). and the total area is two trapezoids, or 24sqrt2. | ||
== See also == | == See also == | ||
{{AMC12 box|year=2006|ab=B|num-b=14|num-a=16}} | {{AMC12 box|year=2006|ab=B|num-b=14|num-a=16}} |
Revision as of 20:26, 29 January 2012
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Contents
Problem
Circles with centers and have radii 2 and 4, respectively, and are externally tangent. Points and are on the circle centered at , and points and are on the circle centered at , such that and are common external tangents to the circles. What is the area of hexagon ?
Solution
Draw the altitude from onto and call the point . Because and are right angles due to being tangent to the circles, and the altitude creates as a right angle. is a rectangle with bisecting . The length is and has a length of , so by pythagorean's, is .
, which is half the area of the hexagon, so the area of the entire hexagon is
Solution 2
ADOP and OPBC are congruent right trapezoids with legs 2 and 4 and with OP equal to 6. Draw an altitude from O to either DP or CP, creating a rectangle with width 2 and base x, and a right triangle with one leg 2, the hypotenuse 6, and the other x. Using the Pythagorean theorem, x is equal to , and x is also equal to the height of the trapezoid. The area of the trapezoid is thus 1/2(4+2)*4sqrt(2) = 12sqrt(2). and the total area is two trapezoids, or 24sqrt2.
See also
2006 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 14 |
Followed by Problem 16 |
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 12 Problems and Solutions |