Question

In the figure above, if the area of triangular region Dis 4, what is the length of a side of square region A ?
(1) The area of square region B is 9.
(2) The area of square region C is $\frac{64}{9}$.

• A. Statement (1) alone is sufficient, but statement (2) alone is not sufficient.
• B. Statement (2) alone is sufficient, but statement (1) alone is not sufficient.
• C. Both statements together are sufficient, but neither statement alone is sufficient.
• D. Each statement alone is sufficient.
• E. Statements (1) and (2) together are not sufficient.

Answer 1

The correct option is D Each statement alone is sufficient.

The area of the triangular region D can be represented by , where b is the base of the triangle (and is equal to the length of a side of the square region C) and h is the height of the triangle (and is equal to the length of a side of the square region B). The area of any square is equal to the length of a side squared. The Pythagorean theorem is used to find the length of a side of a right triangle, when the length of the other 2 sides of the triangle are known and is represented by , where a and b are the lengths of the 2 perpendicular sides of the triangle and c is the length of the hypotenuse. Although completed calculations are provided in what follows, keep in mind that completed calculations are not needed to solve this problem.

1. If the area of B is 9, then the length of each side is 3. Therefore, h = 3. Then, b can be determined, since the area of the triangle is, by substitution, or 8 = 3b. Once b is known, the Pythagorean theorem can be used: . Thus, . The length of a side of A is thus ; SUFFICIENT.
2. If the area of C is , then the length of each side is . Therefore, . The area of the triangle is so ; thus h = 3. Once h is known, the Pythagorean theorem can be used as above; SUFFICIENT. The correct answer is D; each statement alone is sufficient.