a) PS = QR = √37
b) Slope of PS = Slope of QR = -1/6
c) Option A is correct
The figure shown on the coordinate plane is a parallelogram. A parallelogram is a shape with its opposite side parallel and congruent.
From the given diagram, PS = QR and RS = PQ
a) To find the length of PS, we will find the distance between the coordinate points P(1, -3) and S(-5, -2). Using the distance formula;
[tex]\begin{gathered} PS=\sqrt[]{(x_2-x_1)^2+(y_2-y_1)^2} \\ PS=\sqrt[]{(1-(-5))^2+\mleft(-3-\mleft(-2\mright)\mright)} \\ PS=\sqrt[]{(1+5)^2+(-3+2)^2} \\ PS=\sqrt[]{6^2_{}+1^2} \\ PS=\sqrt[]{37} \end{gathered}[/tex]Since PS = QR, hence QR = √37
b) Parallel lines also have the same slope. Therefore, the slope of PS is equal to the slope of QR. Using the slope equation;
[tex]\begin{gathered} m=\frac{y_2-y_1}{x_2-x_1} \\ \text{Slope of PS=}\frac{-2-(-3)}{-5-1} \\ \text{Slope of PS=}\frac{-2+3}{-6} \\ \text{Slope of PS=-}\frac{1}{6} \end{gathered}[/tex]Since the slope of PS is equal to the slope of QR, hence the slope of QR is -1/6
c) As earlier discussed, the opposite sides of a parallelogram are parallel and congruent. Hence we can conclude from the options given that "the quadrilateral is a parallelogram because it has one pair of opposite sides that are both congruent and parallel.
