SAT Math Multiple Choice Question 165: Answer and Explanation

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Question: 165

15. Which of the following lines contains all points equidistant from the points (0, 4) and (8, 0) in the xy-plane?

  • A. 2y = -x + 8
  • B. 2y = x
  • C. y = 2x - 6
  • D. y = -2x

Correct Answer: C

Explanation:

C First, start with a sketch of the two points to see what the line in question might look like.

The point directly between the two points will definitely be on the line, so find the midpoint of the two points. Midpoint = (4, 2). Check this point in the answer choices and eliminate any that do not contain it. Choice (A) becomes 2(2) = -4 + 8 or 4 = 4, which is true. Choice (B) becomes 2(2) = 4, and (C) becomes 2 = 2(4) - 6 or 2 = 8 - 6. These are also true, but (D) becomes 2 = -2(4), which is false. Eliminate (D). To sketch the remaining equations, rewrite them in slope-intercept form of the equation y = mx + b, where m is the slope and b is the y-intercept. Choice (A) becomes y = -x + 4, (B) becomes y = x, and (C) is already in the right form. Now sketch the graphs of each of these on the xy-plane.

The line in (A) contains both the given points, but all the points to the left of (0, 4) are closer to that point and all those to the right of (8, 0) are closer to it. So eliminate (A). Many points on line (B) are also clearly closer to one or the other of the given points, so eliminate (B). Line (C) appears to be perpendicular to the line formed by the two given points, and this is in fact what will make all the points on a line equidistant from 2 given points. Therefore, the correct answer is (C).

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