Quadrilateral PQRS is not a rectangle because analysis of the slopes of its sides reveals that none of the angles in the quadrilateral are right angles.
In addressing whether quadrilateral PQRS with vertices P(−4, 2), Q(3, 4), R(5, 0), and S(−3, −2) is a rectangle, one must consider the properties that define a rectangle.
A rectangle is a quadrilateral with four right angles.
If all four angles are not right angles, the quadrilateral cannot be a rectangle.
We can analyze the slopes of the sides to determine if there are any right angles.
The slope between two points (x1, y1) and (x2, y2) is given by (y2 - y1) / (x2 - x1).
For sides PQ, QR, RS, and SP, we calculate the slopes as follows:
Slope of PQ: (4 - 2) / (3 + 4) = 2/7
Slope of QR: (0 - 4) / (5 - 3) = -4/2 = -2
Slope of RS: (-2 - 0) / (-3 - 5) = -2/(-8) = 1/4
Slope of SP: (2 + 2) / (-4 + 3) = 4/(-1) = -4
Observing the slopes, we see no pairs of perpendicular slopes (opposite reciprocals).
Hence, there are no right angles, and quadrilateral PQRS is not a rectangle.
The correct statement describing quadrilateral PQRS is option b, Quadrilateral PQRS is not a rectangle because it has no right angles.
The probable question may be:
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?
a. Quadrilateral PQRS is not a rectangle because it has only one right angle.
b. Quadrilateral PQRS is not a rectangle because it has no right angles.
c. Quadrilateral PQRS is not a rectangle because it has only two right angles.
d. Quadrilateral PQRS is a rectangle because it has four right angles.