6.2.2 Conditions for Special Parallelograms (continued) Example 3 Identifying Special Parallelograms in the Coordinate Plane
The properties of the diagonals of a parallelogram in the coordinate plane are used here to determine whether the parallelogram is a rectangle, a rhombus, or a square. The coordinates of the parallelogram are given. First, test whether the parallelogram is a rectangle by determining if the diagonals are the same length. Calculate the length of the diagonals using the Pythagorean Theorem. The diagonals are congruent, so the figure is a rectangle by the conditions for rectangles. Second, test whether the parallelogram is a rhombus by determining if the slopes of the diagonals are perpendicular. The slopes of the diagonals are 3/7 and − 7/3, and the product of the slopes is − 1, so the diagonals are perpendicular and the figure is a rhombus by the conditions for a rhombus. The parallelogram is also a square because it is both a rectangle (four right angles) and a rhombus (four congruent sides).
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