Question

1. the vertices of quadrilateral abcd are given. draw abcd in a coordinate plane and show that it is a parallelogram. a(-1,3), b(4,3), c(2,-1), d(-3,-1)

Explanation:

Step1: Calculate the slope of \(AB\)

The slope formula is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). For points \(A(-1,3)\) and \(B(4,3)\), \(m_{AB}=\frac{3 - 3}{4-(-1)}=\frac{0}{5}=0\)

Step2: Calculate the slope of \(CD\)

For points \(C(2,-1)\) and \(D(-3,-1)\), \(m_{CD}=\frac{-1-(-1)}{-3 - 2}=\frac{0}{-5}=0\)

Step3: Calculate the slope of \(BC\)

For points \(B(4,3)\) and \(C(2,-1)\), \(m_{BC}=\frac{-1 - 3}{2-4}=\frac{-4}{-2}=2\)

Step4: Calculate the slope of \(DA\)

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

Since \(m_{AB}=m_{CD}\) and \(m_{BC}=m_{DA}\), opposite sides are parallel.

Answer:

Since the slopes of opposite sides \(AB\) and \(CD\) are equal (\(m_{AB}=m_{CD} = 0\)), and the slopes of opposite sides \(BC\) and \(DA\) are equal (\(m_{BC}=m_{DA}=2\)), quadrilateral \(ABCD\) is a parallelogram.