Question

The vertices of quadrilateral PQRS are P(4, 5), Q(6, 6), R(9, 4), and S(7, 2). To determine whether quadrilateral PQRS is a parallelogram, we can use the slope criteria. If the slopes of opposite sides are equal, then the quadrilateral is a parallelogram. Let's denote the coordinates as P(x₁, y₁), Q(x₂, y₂), R(x₃, y₃), and S(x₄, y₄). The slope of a line passing through two points (x₁, y₁) and (x₂, y₂) is given by (y₂ - y₁)/(x₂ - x₁). Calculate the slopes of PQ, QR, RS, and SP. If the slopes of PQ and RS are equal, and the slopes of QR and SP are equal, then the quadrilateral is a parallelogram.

Answer 1

To find out if PQRS is a parallelogram, the slopes of opposite sides are calculated. The slopes of PQ and RS are both 1/2, and the slopes of QR and SP are both -2/3. Since opposite sides have equal slopes, PQRS is a parallelogram.

Step-by-step explanation:

To determine if quadrilateral PQRS with vertices P(4, 5), Q(6, 6), R(9, 4), and S(7, 2) is a parallelogram, we have to calculate the slopes of the sides. The slope of a line passing through two points (x₁, y₁) and (x₂, y₂) is (y₂ - y₁)/(x₂ - x₁).

Calculating the slopes:

SP slope = (5 - 2)/(4 - 7) = -2/3

The slopes of PQ and RS are equal, and the slopes of QR and SP are equal, which implies that opposite sides are parallel and that PQRS is a parallelogram according to the slope criteria.