A segment AB is given. Find the coordinates A if B (-2; 3; 4) and the midpoint of the segment O (5; -2; 3).

1. Given the coordinates of one of the ends of the segment AB and its midpoint O. It is necessary to find the coordinates of the second end:

B (-2; 3; 4);
O (5; -2; 3);
A =?
2. The coordinates of the midpoint of the segment are the arithmetic mean of the corresponding coordinates of its ends:

x (O) = (x (A) + x (B)) / 2; (one)
y (O) = (y (A) + y (B)) / 2; (2)
z (O) = (z (A) + z (B)) / 2. (3)
3. Having solved equations (1), (2) and (3) with respect to x (A), y (A) and z (A), we find the coordinates of point A:

x (A) = 2x (O) – x (B) = 2 * 5 + 2 = 12;
y (A) = 2y (O) – y (B) = 2 * (-2) – 3 = -7;
z (A) = 2z (O) – z (B) = 2 * 3 – 4 = 2;
A (12; -7; 2).
Answer: A (12; -7; 2).