Question

3. Triangle XYZ is inscribed in semicircle O shown below such that yZ = 7 and YX = 10. Which of the following is closest to the radius of the semicircle? (1) 5.8 (2) 6.1 (3) 11.6 (4) 12.2

Answer 1

Using the Pythagorean theorem, the hypotenuse of right triangle XYZ is equal to the diameter of the semicircle. The hypotenuse is the square root of the sum of the squares of the other two sides, leading us to an approximate radius of 6.1 units, which is option (2).

Step-by-step explanation:

We are looking to find the radius of the semicircle in which triangle XYZ is inscribed. Since the triangle is inscribed in a semicircle, we can use the fact that the hypotenuse of the right triangle formed (triangle XYZ in this case) will be the diameter of the semicircle. To find the hypotenuse, we apply the Pythagorean theorem to sides YZ and YX:

YZ2 + YX2 = XY2

72 + 102 = XY2

49 + 100 = XY2

XY2 = 149

XY = √149

Since XY is the diameter, the radius (r) is half the diameter. Thus, the radius is:

r = XY / 2

r = √149 / 2

r ≈ 6.1 (rounding to the nearest tenth)

Therefore, the closest answer to the radius of the semicircle is option (2) 6.1.