Question

1. you are designing a garden with four corners represented by the points p(-2, 1), q(3, 4), r(5, 1), and s(0, -2). will the shape of your garden be a parallelogram?

Explanation:

Step1: Recall the property of a parallelogram (opposite sides are equal and parallel, so their slopes are equal)

To find the slope between two points \((x_1,y_1)\) and \((x_2,y_2)\), we use the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\)

Step2: Calculate the slope of \(PQ\)

For points \(P(-2,1)\) and \(Q(3,4)\), the slope \(m_{PQ}=\frac{4 - 1}{3 - (-2)}=\frac{3}{5}\)

Step3: Calculate the slope of \(SR\)

For points \(S(0,-2)\) and \(R(5,1)\), the slope \(m_{SR}=\frac{1 - (-2)}{5 - 0}=\frac{3}{5}\)

Step4: Calculate the slope of \(QR\)

For points \(Q(3,4)\) and \(R(5,1)\), the slope \(m_{QR}=\frac{1 - 4}{5 - 3}=\frac{- 3}{2}\)

Step5: Calculate the slope of \(PS\)

For points \(P(-2,1)\) and \(S(0,-2)\), the slope \(m_{PS}=\frac{-2 - 1}{0 - (-2)}=\frac{-3}{2}\)

Since \(m_{PQ}=m_{SR}\) and \(m_{QR}=m_{PS}\), the opposite sides are parallel. So the quadrilateral \(PQRS\) is a parallelogram.

Answer:

Yes, the shape of the garden will be a parallelogram.