2005 AMC 8 Problems/Problem 23
Problem
Isosceles right triangle encloses a semicircle of area . The circle has its center on hypotenuse and is tangent to sides and . What is the area of triangle ?
Solution
The semi circle has an area of and a radius of .
Because this is an isosceles right triangle, the center is the midpoint of the hypotenuse. Radii drawn to the tangent points of the semicircle and the radii also divide the legs into two equal segments. They also create a square in the top left corner. From this, we can conclude the legs of the triangle are twice the length of the radii, . The area of the triangle is .
Easier and More Logical Solution
First, we notice half a square so first let's create a square. Once we have a square, we will have a full circle. This circle has a diameter of 4 which will be the side of the square. The area would be 4*4 = 16. Divide 16 by 2 to get the original shape and you get
See Also
2005 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 22 |
Followed by Problem 24 | |
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