If in the figure above, semicircles X and Y have radii of 6 and 8 feet, respectively, then what is the area of semicircle Z (in square feet)? (Note: Figure not drawn to scale.) A. 10π B. 20π C. 25π D. 50π E. 100π
(D) If the radii of the semicircles X and Y are 6 feet and 8 feet, then their diameters are 12 and 16 feet. These diameters are both lengths of the legs of a right triangle, and the diameter of semicircle Z is the hypotenuse of this triangle.
The triangle is a 3-4-5 triangle, since the perpendicular legs are in the ratio 3 : 4. So the hypotenuse measures 20 feet, meaning the diameter of Z is 20 feet and its radius is 10 feet.
To determine the area of semicircle Z, we will determine the area of a full circle with the same radius as Z, and then divide this quantity by 2.
The area of a circle with radius 10 is: Area = πr² Area = π10² = 100π.
Thus, the area of semicircle Z is 50π square feet. The correct answer is choice (D). ----------
If the triangle is a 3:4:5,then shouldn't the legs of the triangle be 12, 16 and 25? The hypotenuse shouldn't be 20, it should be 25, correct? This means the radius is 25/2.
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