Square ABCD is inscribed in circle O

David Recine

The official answer from the guide is D: either statement is sufficient on its own. Let's take a look at why. The two statements are:

(1) The area of circular region O is 64.
(2) The circumference of circle O is 16.

With statement 1, we have the area of the circle as our starting point. We also know that (r^2)= the area of a circle. And 8^2 is 64. So, r (the radius) for the circle must be 8. And of course, diameter is 2r, so this statement gets us the diameter of the circle: 16.

Now that statement 1 has gotten us 16 for the diameter of the circle, we also have the diagonal of the square. Since the square is inscribed, the diagonal of the square will bisect both the circle and the square, and this the diameter of the circle and the diagonal of the square will be the same. So, the diagonal of the square is 16.

From this diagonal of 16 derived from Statement 1, you can get the area of the square. How? By realizing that a bisected square is the same thing as two right triangles with equal sides around the right angle. From there, treat the equal sides as a and b, where a=b, and treat the diagonal of the square as side c. Next, apply the Pythagorean theorem: a^2+b^2= c^2, or in this case, a^2+b^2 = 16^2 >>> a^2+b^2 = 256.

Now remember, sides a and b are equal, and are also the sides of the square, the sides you actually need a measurement on in order to find the area of the square. So we can change our equation from a^2+b^2 = 256 to s^2+s^2 = 256 and solve for s as follows:
2(s^2)= 256
[2(s^2)]/2 = 256/2
s^2 = 128
At this point, you know that each side of the square equals the square root of 128, and that s*s = the area of the square. The square root of 128, squared, is 128, so that's the area of the square. SUFFICIENT

Now, let's look at statement 2. This says that the circumference of the circle is 16. Now, the formula for circumference is 2r. So 2r= 16. Divide both sides by , and we have 2r = 16. this gives us an r value of 8. And, through the steps seen above for Statement 1, if you have r, you can get diameter of the circle and diagonal of the square. As with statement 1, this in turn allows you to use the Pythagorean theorem to get the length of the sides of the square, which at long last leads to the area of the square. SUFFICIENT.

Dec 27, 2017 • Comment

Mich Greenberg

Magoosh team - could you kindly provide an answer to this question?

Dec 13, 2017 • Comment