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.

If the triangle is a 3:4:5,then shouldn't the legs of the triangle be 12, 16 and 25?

No, your reasoning is NOT correct.

The 12 : 16 : 25 triangle is a 3 : 4 : 25/4, but NOT a 3 : 4 : 5 triangle. You can also check, that 25² = 625, while 12² + 16² = 144 + 256 = 400 = 20².

So note, that a 3:4:5 triangle is a triangle, which sides result in a 3:4:5 proportion, which is the same as 3a : 4a : 5a.

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