Question

Determine whether a figure with the given vertices is a parallelogram. Use the method indicated.

Answer 1

Answer with explanation:

The coordinates of four points of Quadrilateral ABCD are

    A(4, 5), B(–7, –10), C(–4, –9), D(7, 6)

$AB=\sqrt{(4+7)^2+(5+10)^2}\\\\AB=\sqrt{121+225}\\\\AB=\sqrt{346}\\\\BC=\sqrt{(-7+4)^2+(-10+9)^2}\\\\BC=\sqrt{9+1}\\\\BC=\sqrt{10}\\\\CD=\sqrt{(4+7)^2+(5+10)^2}\\\\CD=\sqrt{121+225}\\\\CD=\sqrt{346}\\\\DA=\sqrt{(7-4)^2+(6-5)^2}\\\\DA=\sqrt{9+1}\\\\DA=\sqrt{10}$

Slope between two points is given by

 $m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\\\\m_{1}=\frac{-10-5}{-7-4}\\\\=\frac{-15}{-11}\\\\m_{1}=\frac{15}{11}\\\\m_{2}=\frac{-9+10}{-4+7}\\\\m_{2}=\frac{1}{3}\\\\m_{3}=\frac{6+9}{7+4}\\\\m_{3}=\frac{15}{11}\\\\m_{4}=\frac{6-5}{7-4}\\\\m_{4}=\frac{1}{3}$

As slope of opposite sides are equal and length of Opposite sides are equal.So,the given Quadrilateral is a Parallelogram.

Option A

Yes; The opposite sides are congruent and have the same slope.