Question

quadrilateral pqrs is given by these coordinates: p(-10,6), q(-8,8), r(-4,6), and s(-8,2).
jack claims that quadrilateral pqrs is a parallelogram.
select the correct statement about jacks claim.
a. jack is incorrect. the figure is not a parallelogram because opposite sides are not congruent.
b. jack is incorrect. the figure is not a parallelogram because all sides are not congruent.
c. jack is correct. the figure is a parallelogram because a pair of opposite sides is parallel.
d. jack is correct. the figure is a parallelogram because the angles are not all right angles.

Explanation:

Step1: Calculate the lengths of the sides

Use the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
For \(PQ\): \(x_1=-10,y_1 = 6,x_2=-8,y_2 = 8\), \(d_{PQ}=\sqrt{(-8+10)^2+(8 - 6)^2}=\sqrt{4 + 4}=\sqrt{8}=2\sqrt{2}\).
For \(QR\): \(x_1=-8,y_1 = 8,x_2=-4,y_2 = 6\), \(d_{QR}=\sqrt{(-4 + 8)^2+(6 - 8)^2}=\sqrt{16+4}=\sqrt{20}=2\sqrt{5}\).
For \(RS\): \(x_1=-4,y_1 = 6,x_2=-8,y_2 = 2\), \(d_{RS}=\sqrt{(-8 + 4)^2+(2 - 6)^2}=\sqrt{16 + 16}=\sqrt{32}=4\sqrt{2}\).
For \(SP\): \(x_1=-8,y_1 = 2,x_2=-10,y_2 = 6\), \(d_{SP}=\sqrt{(-10+8)^2+(6 - 2)^2}=\sqrt{4+16}=\sqrt{20}=2\sqrt{5}\).

Step2: Check the properties of a parallelogram

In a parallelogram, opposite sides are congruent. \(PQ = RS=2\sqrt{2}\) and \(QR=SP = 2\sqrt{5}\). Also, the slope of \(PQ\): \(m_{PQ}=\frac{8 - 6}{-8+10}=1\), the slope of \(QR\): \(m_{QR}=\frac{6 - 8}{-4 + 8}=-\frac{1}{2}\), the slope of \(RS\): \(m_{RS}=\frac{2 - 6}{-8+4}=1\), the slope of \(SP\): \(m_{SP}=\frac{6 - 2}{-10 + 8}=-2\). Since \(m_{PQ}=m_{RS}\) and \(m_{QR}=m_{SP}\), opposite sides are parallel.

Step3: Analyze Jack's claim

Jack claims it is a parallelogram. A parallelogram does not require all sides to be congruent (that's a rhombus) or all angles to be right angles (that's a rectangle). The key property of a parallelogram is that opposite sides are parallel and congruent.

Answer:

D. Jack is correct. The figure is a parallelogram because the angles are not all right angles.