The lengths of three sides of a triangle are x, x + 2, and x + 4. What is the area of the triangle?
(1) The triangle is a right triangle. (2) The lengths of two of the sides (in inches) are 8 and 10.
A. Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by itself is not. B. Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by itself is not. C. Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, even though NEITHER statement BY ITSELF is sufficient. D. Either statement BY ITSELF is sufficient to answer the question. E. Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question, meaning that further information would be needed to answer the question.
(A) Statement (1) tells us that the Pythagorean theorem will hold for this triangle. x + 4 must be the length of the hypotenuse, since it is the largest number. Using the Pythagorean theorem, we can solve for x: x² + (x + 2)² = (x + 4)² x² + x² + 4x + 4 = x² + 8x + 16 x² = 4x + 12 x² – 4x – 12 = 0 (x + 2)(x – 6) = 0 x = 6 or -2.
We know that x cannot be negative, since the lengths of all sides must be positive. Therefore, x = 6 and the other sides of the triangle must be 8 and 10. So we can determine the area. While it is not necessary to do so, we could calculate the area as follows: Area = (1/2) × 6 × 8 = 24.
So Statement (1) is sufficient.
Statement (2) tells us the lengths of two sides, but we do not know the third. 8 and 10 can be either the two smaller or two larger sides, so the side lengths could either be 6, 8, and 10 or 8, 10, and 12. These two triangles have different areas. So Statement (2) is insufficient.
Since Statement (1) is sufficient and Statement (2) is insufficient, the correct answer is choice (A).\ ------------- Hi, I think the answer for this should be D. Statement (2) is also sufficient, if we know 8 and 10 are two sides then we find the third side using Pythagorean theorem we can't get 12 as an answer because it will contradict with premise given x, x+2 and x+4. If 8 and 10 are smaller side then hypotenus would be 12.81 hence not x + 4. only option is 6.
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