Three vertices of the parallelogram ABCD are given. Find its fourth vertex D if A (1; 3) B (2; 6) C (-3; 1)

By the property of the lengths of the sides of the parallelogram: AC = BD and these sides are parallel.

Hence, to find the coordinates of the point D, it is necessary to find the length of the segment AC and equate it to BD. Formula for finding the length of a segment:

(x2 – x1) ² + (y2 – y1) ² = d²;

where:

d is the length of the segment.

x, y – corresponding coordinates.

AC length:

AC² = (-3 – 1) 2 + (1 – 3) 2 = 20;

AC = √20;

Side length BD:

BD² = (x2 – 2) ² + (y2 – 6) ²;

(x2 – 2) ² + (y2 – 6) ² = 20

From the condition of parallelism of the sides, we obtain that x2 = -2, and y2 = 4.

Answer: x2 = -2, y2 = 4.