Question

1. show that quadrilateral abcd is a parallelogram using the definition of parallelogram: show that both pairs of opposite sides are parallel. a(-3,2), b(-2,7), c(2,4), & d(1,-1)

Explanation:

Step1: Calculate the slope of \(AB\)

The slope formula is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). For points \(A(-3,2)\) and \(B(-2,7)\), \(m_{AB}=\frac{7 - 2}{-2-(-3)}=\frac{5}{1} = 5\)

Step2: Calculate the slope of \(CD\)

For points \(C(2,4)\) and \(D(1,-1)\), \(m_{CD}=\frac{-1 - 4}{1 - 2}=\frac{-5}{-1}=5\)

Step3: Calculate the slope of \(BC\)

For points \(B(-2,7)\) and \(C(2,4)\), \(m_{BC}=\frac{4 - 7}{2-(-2)}=\frac{-3}{4}\)

Step4: Calculate the slope of \(DA\)

For points \(D(1,-1)\) and \(A(-3,2)\), \(m_{DA}=\frac{2-(-1)}{-3 - 1}=\frac{3}{-4}=-\frac{3}{4}\)

Since \(m_{AB}=m_{CD} = 5\) and \(m_{BC}=m_{DA}=-\frac{3}{4}\), both pairs of opposite sides are parallel.

Answer:

Quadrilateral \(ABCD\) is a parallelogram as \(AB\parallel CD\) (slope \( = 5\)) and \(BC\parallel DA\) (slope \(=-\frac{3}{4}\))