2006 AMC 10B Problems/Problem 24
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 ?
Video Solution 1
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Solution 1 (Similar Triangles)
When we see this problem, it practically screams similar triangles at us. Extend to the left until it intersects lines and at point . Triangles and are similar, and by symmetry, so are triangles and . Then, the area of kite is just and the area of kite is (using our similarity ratios). The difference of these yields the area of hexagon or .
Solution 2 (Perpendiculars)
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 .
Note: Quadrilaterals and are congruent, so they have equal areas.
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