SAT Math Multiple Choice Practice Question 586: Answer and Explanation
22. If a square lies completely within a circle, which of the following must be true?
I. The radius of the circle is equal in length to one side of the square.
II. The area of the square is less than the area of the circle.
III. All four corners of the square touch the circle.
A. I only
B. II only
C. I and II only
D. II and III only
E. I, II, and III
Correct Answer: B
B Draw a figure if one is described in the question but not shown. In this case, we know a square lies in a circle, and the way we draw these shapes depends on what is in each of the statements above the answers. Since the question wants to know what must be true, try to draw a figure to disprove each statement. Statement I can be disproved with a drawing such as
Eliminate A, C, and E because they all contain I. B and D both contain II, so it must be true. Check out statement III. The circle above disproves III, so get rid of D. Only B can be correct. Skipping statement II saves time: To prove it true you'd need to make the biggest possible square inside the circle, such as
The diameter of the circle is the same as the diagonal of the square. If the diameter were 10, then the radius would be 5, and the area of the circle would be 25π. To find the area of the square, we can use the diagonal as the hypotenuse of a 45-45-90 triangle. The legs and hypotenuse have a ratio of x : x : x. In this case, the value of 10 = x. Solve for x to find the side of the square: . The area of the square is . This is less than 25π, which is a little over 75.