This is an inscribed right triangle and circle circumference problem.

What is known about right triangles? The relationship between the hypotenuse (c) and the two legs (a and b) is given by the pythagorean theorem a^2 + b^2 = c^2.

What is known about circles and circumferences? C (circumference) = pi*d where d is the diameter.

The pictured situation is special. There's something else known. When the end points of an inscribed angle (angle on the edge of the circle) touch the opposite ends of the circle's diameter, the angle is a right angle. Angle wxy is a right angle. xz is an altitude from the right angle (wxy) perpendicular to the hypotenuse (wy). This creates similar right triangles. The big right triangle is similar to both of the smaller right triangles (and they are similar to each other). The ratio of the hypotenuse to the long or short leg of each triangle will be the same. HYP/LONG for the big triangle = hyp/long for the smaller triangles. Also HYP/SHORT = hyp/short.

hyp/long: wy/xw = xy/xz = xw/wz and hyp/short: wy/xy = xy/yz = xw/xz


Anticipate the information necessary to answer the question:

If the statement provides the diameter, we can find circumference.

If the statement(s) provides the length of any other side of either of the two small triangle, then we can find the third side, and use the ratios (shown above) to find the diameter of the circle. The diameter gives the circumference (pi*d)

It's not necessary to calculate the numbers. Knowing you can is sufficient.

1. wx is 40. wx is the long leg of triangle wxy and the hypotenuse of wzx. xz is known. xz is the short leg of wzx. Use pyhagorean theorem to find the long leg of wzx, then use the ratio hyp/long = HYP/long to find the HYP of wxy, which is the diameter of the circle. Use the diameter to find the circumference. Sufficient

Eliminate B, C, and E. A and D are the possible answers.


2. xy = 30. xy is the short leg of wxy, and the hypotenuse of xzy. xz is the long leg of xzy. Use the pythagorean theorem to find the short leg of xzy. Then use HYP/SHORT = hyp/short to find the HYP (wy), which is diameter. Use diameter to find the circumference. Sufficient.

Eliminate A. The answer is D.