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 649.
A
Statement (1) alone is sufficient, but statement (2) alone is not sufficient.
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B
Statement (2) alone is sufficient, but statement (1) alone is not sufficient.
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C
Both statements together are sufficient, but neither statement alone is sufficient.
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D
Each statement alone is sufficient.
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E
Statements (1) and (2) together are not sufficient.
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Solution
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.
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.
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.