Find a point M equidistant from three points A (2; 2), B (-5; 1), C (3, -5).

| July 28, 2021 | education

Consider the Cartesian coordinate system Oxy and on it the points A (2; 2), B (-5; 1) and C (3, -5). At the request of the task, we find the coordinates of the point M, equidistant from the given three points A, B and C. Let’s denote the coordinates of the point M through x and y.
Since point M is equidistant from points A, B and C, the equalities AM = BM = CM are fulfilled. Using the formula for calculating the distance between two points, we have: AM = √ ((x – 2) ² + (y – 2) ²), BM = √ ((x – (-5)) ² + (y – 1) ²) and AM = √ ((x – 3) ² + (y – (-5)) ²). Squaring both sides of each equality, we get: (x – 2) ² + (y – 2) ² = (x + 5) ² + (y – 1) ² = (x – 3) ² + (y + 5) ².
Let’s open the brackets and simplify: x² – 4 * x + 4 + y² – 4 * y + 4 = x² + 10 * x + 25 + y² – 2 * y + 1 = x² – 6 * x + 9 + y² + 10 * y + 25 or -2 * x – 2 * y = 5 * x – y + 13 = -3 * x + 5 * y + 17.
From the first equality, we get y = -7 * x – 9, and from the last: 3 * y = 4 * x – 2. Then 3 * (-7 * x – 9) = 4 * x – 2, whence x = – one. Therefore, y = -7 * (-1) – 9 = -2.
Answer: M (-1; -2).



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