Question

The points W, left bracket, minus, 3, comma, minus, 3, right bracket, comma, X, left bracket, 0, comma, minus, 5, right bracket, comma, Y, left bracket, 7, comma, 0, right bracketW(−3,−3),X(0,−5),Y(7,0), and Z, left bracket, 4, comma, 2, right bracketZ(4,2) form quadrilateral WXYZ.. I will prove that quadrilateral WXYZ is a parallelogram by demonstrating that:

Answer 1

1. Opposite sides of WXYZ are parallel:
We can prove this by calculating the slopes of the line segments connecting the opposite vertices. The slope of line segment WX is:

m(WX) = (Y2 - Y1) / (X2 - X1)
= (0 - (-3)) / (7 - 0)
= 3/7

The slope of line segment YZ is:

m(YZ) = (Y2 - Y1) / (X2 - X1)
= (2 - 0) / (4 - 7)
= -2/3

Since the slopes are not equal, m(WX) ≠ m(YZ), line segments WX and YZ are not parallel.

The slope of line segment WZ is:
m(WZ) = (Y2 - Y1) / (X2 - X1)
= (2 - (-3)) / (4 - (-3))
= 5/7

The slope of line segment XY is:
m(XY) = (Y2 - Y1) / (X2 - X1)
= (0 - (-5)) / (7 - 0)
= 5/7

Since the slopes are equal, m(WZ) = m(XY), line segments WZ and XY are parallel.

2. Opposite sides of WXYZ are congruent:
We can calculate the lengths of the line segments to prove this.

Length of WX:
√[(X2 - X1)^2 + (Y2 - Y1)^2]
= √[(0 - (-3))^2 + (-5 - (-3))^2]
= √[3^2 + (-5 + 3)^2]
= √[9 + 4]
= √13

Length of YZ:
√[(X2 - X1)^2 + (Y2 - Y1)^2]
= √[(4 - 7)^2 + (2 - 0)^2]
= √[(-3)^2 + 2^2]
= √[9 + 4]
= √13

Length of WZ:
√[(X2 - X1)^2 + (Y2 - Y1)^2]
= √[(4 - (-3))^2 + (2 - (-3))^2]
= √[7^2 + 5^2]
= √[49 + 25]
= √74

Length of XY:
√[(X2 - X1)^2 + (Y2 - Y1)^2]
= √[(0 - 7)^2 + (-5 - 0)^2]
= √[(-7)^2 + (-5)^2]
= √[49 + 25]
= √74

Since the lengths are equal, WX ≅ YZ and WZ ≅ XY.

3. Diagonals of WXYZ bisect each other:
We can show this by calculating the midpoints of the diagonals. The midpoint of WZ is:

((X1 + X2) / 2, (Y1 + Y2) / 2)
= ((-3 + 4) / 2, (-3 + 2) / 2)
= (1/2, -1/2)

The midpoint of XY is:

((X1 + X2) / 2, (Y1 + Y2) / 2)
= ((0 + 7) / 2, (-5 + 0) / 2)
= (7/2, -5/2)

The coordinates of the midpoints, (1/2, -1/2) and (7/2, -5/2), are equal.

Therefore, quadrilateral WXYZ satisfies all the conditions of a parallelogram.