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Re: In the rectangular coordinate system shown above, points O, P, and Q r
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08 Dec 2020, 11:10
OFFICIAL GMAT EXPLANATION
Geometry Coordinate geometry
Any point equidistant from the points (0,0) and (4,0) must lie on the perpendicular bisector of the segment with endpoints (0,0) and (4,0), which is the line with equation x = 2. Any point equidistant from the points (0,0) and (0,6) must lie on the perpendicular bisector of the segment with endpoints (0,0) and (0,6), which is the line with equation y = 3. Therefore, the point that is equidistant from (0,0), (4,0), and (0,6) must lie on both of the lines x = 2 and y = 3, which is the point (2,3).
Alternatively, let (x, y) be the point equidistant from (0,0), (4,0), and (0,6). Since the distance between (x, y) and (0,0) is equal to the distance between (x, y) and (4,0), it follows from the distance formula that √x^2 + y^2 = √(x−4)^2 + y^2. Squaring both sides gives x^2 + y^2 = (x − 4)^2 + y^2. Subtracting y^2 from both sides of the last equation and then expanding the right side gives x^2 = x^2 − 8x + 16, or 0 = −8x + 16, or x = 2. Also, since the distance between (x, y) and (0,0) is equal to the distance between (x, y) and (0,6), it follows from the distance formula that √x^2 + y^2 = √x^2 + (y−6)^2. Squaring both sides of the last equation gives x^2 + y^2 = x^2 + (y − 6)^2. Subtracting x^2 from both sides and then expanding the right side gives y^2 = y^2 − 12y + 36, or 0 = −12y + 36, or y = 3.