Question

The coordinates of the vertices of quadrilateral PQRS are P(−4, 2), Q(3, 4), R(5, 0), and S(−3, −2). Which statement correctly describes whether quadrilateral PQRS is a rectangle?

Answer 1

To determine if quadrilateral PQRS is a rectangle, we need to check if opposite sides are parallel and have equal lengths. Checking the slopes of the sides shows that opposite sides are not parallel. Calculating the lengths of the sides reveals that opposite sides do not have equal lengths. Therefore, quadrilateral PQRS is not a rectangle.

Step-by-step explanation:

To determine if quadrilateral PQRS is a rectangle, we need to check if opposite sides are parallel and if opposite sides have equal lengths.

1. To check if opposite sides are parallel, we can calculate the slopes of the lines connecting the pairs of opposite vertices: PQ, QR, RS, and SP.

• The slope of PQ is (4-2)/(3-(-4)) = 2/7.
• The slope of QR is (0-4)/(5-3) = -2/2 = -1.
• The slope of RS is (-2-0)/(-3-5) = -2/(-8) = 1/4.
• The slope of SP is (2-(-2))/(-4-(-3)) = 4/-1 = -4.

The slopes of PQ and RS are not equal, and the slopes of QR and SP are not equal, so opposite sides are not parallel. Therefore, quadrilateral PQRS is not a rectangle.

2. To check if opposite sides have equal lengths, we can calculate the lengths of the sides PQ, QR, RS, and SP using the distance formula:

• The length of PQ is sqrt((3-(-4))^2 + (4-2)^2) = sqrt(7^2 + 2^2) = sqrt(53).
• The length of QR is sqrt((5-3)^2 + (0-4)^2) = sqrt(2^2 + (-4)^2) = sqrt(20).
• The length of RS is sqrt((-3-5)^2 + (-2-0)^2) = sqrt((-8)^2 + (-2)^2) = sqrt(68).
• The length of SP is sqrt((-4-(-3))^2 + (2-(-2))^2) = sqrt((-1)^2 + 4^2) = sqrt(17).

Since the lengths of the opposite sides are not equal, quadrilateral PQRS is not a rectangle.