Quadrilateral ABCD has vertices of points A (-2; -1) B (-3; 2) C (3; 2) D (4; -1). Prove that ABCD

| September 20, 2021 | education

Quadrilateral ABCD has vertices of points A (-2; -1) B (-3; 2) C (3; 2) D (4; -1). Prove that ABCD is a parallelogram and find the diagonal AC.

For the solution, we use the formula for determining the length of a segment by its coordinates.

d = √ (X2 – X1) ² + (Y2 – Y1) ², where d is the length of the segment, X1, X2, Y1, Y2 are the coordinates of the points of the segment.

AB √ (-3 – (-2)) ² + (2 – (-1)) ² = √1 + 9 = √10 cm.

СD = √ (4 – 3) ² + (-1 – 2) ² = √1 + 9 = √10 cm.

AD = √ (4 – (-2)) ² + (-1 – (-1)) ² = √36 + 0 = √36 = 6 cm.

ВС = √ (3 – (-3)) ² + (2 – (-2)) ² = √36 + 0 = √36 = 6 cm.

The lengths of the opposite sides of the quadrangle are equal, it can be a rectangle or a parallelogram. Let us determine the lengths of its diagonals.

AC = √ (3 – (-2)) ² + (2 – (-1)) ² = √25 + 9 = √34 cm.

ВD = √ (4 – (-3)) ² + (-1 – (-2)) ² = √49 + 1 = √50 cm.

Since the diagonals have different lengths, ABCD is a parallelogram, as required.

Answer: The length of the AC diagonal is √34 cm.



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