To classify the quadrilateral find the slopes of the sides. You can also find the lengths of the sides.
Rhombus
Practice makes perfect
We will begin by identifying the coordinates of the vertices in the given parallelogram.
Next, we will find the coordinates of the side's midpoints of our parallelogram. Then we will draw them on the coordinate plane and classify the formed quadrilateral.
Finding the Coordinates of the Midpoints
To find the coordinates of the midpoints, we will substitute the coordinates of the endpoints of each side in the Midpoint Formula.
Side
Midpoint Formula
Simplified
Reference Name
Midpoint of F( - 2,1) and E( 0,2)
( - 2+ 0/2, 1+ 2/2 )
(- 1,1.5)
A
Midpoint of E( 0,2) and H( 2, -2)
( 0+ 2/2,2+( -2)/2 )
(1,0)
B
Midpoint of H( 2, -2) and G( 0, - 3)
(2+ 0/2, - 2+( - 3)/2 )
(1,- 2.5)
C
Midpoint of G( 0, - 3), and F( - 2,1)
( 0+( - 2)/2, - 3+ 1/2 )
(- 1,- 1)
D
Drawing the Quadrilateral
To draw the quadrilateral we will plot the midpoints on the coordinate plane and connect them.
Classifying the Quadrilateral
Let's review the classification of quadrilaterals.
Quadrilateral with two pairs of consecutive sides congruent and no opposite sides congruent.
To classify our quadrilateral, we will begin by finding the slopes of its sides using the Slope Formula.
Side
Slope Formula
Simplified
Slope of DA: ( - 1,- 1), ( - 1,1.5)
1.5-( - 1)/- 1-( - 1)
undefined
Slope of AB: ( - 1,1.5), ( 1, 0)
0- 1.5/1-( - 1)
- 3/4
Slope of BC: ( 1, 0), ( 1, - 2.5)
- 2.5- 0/1- 1
undefined
Slope of CD: ( 1, - 2.5), ( - 1,- 1)
- 1-( - 2.5)/- 1- 1
- 3/4
We see that the slopes of the opposite sides of our quadrilateral are equal. Therefore, both pairs of opposite sides are parallel and our quadrilateral is a parallelogram. Additionally, the consecutive sides are not perpendicular, because their slopes are not opposite reciprocals. Therefore, our quadrilateral can only be a rhombus. To check, we can find the lengths of its sides using the Distance Formula.
Side
Distance Formula
Simplified
Length of DA: ( - 1,- 1), ( - 1,1.5)
sqrt(( - 1-( - 1))^2+( 1.5-( - 1))^2)
6.25
Length of AB: ( - 1,1.5), ( 1, 0)
sqrt(( 1-( - 1))^2+( 0- 1.5)^2)
6.25
Length of BC: ( 1, 0), ( 1, - 2.5)
sqrt(( 1- 1)^2+( - 2.5- 0)^2)
6.25
Length of CD: ( 1, - 2.5), ( - 1,- 1)
sqrt(( -4-( -2))^2+( -2-( -3))^2)
6.25
As we can see, our quadrilateral has four congruent sides. Therefore, it is a rhombus.
