Question

The vertices of quadrilateral DEFG are D 3, 2 ( ) , E 7, 4 ( ), F 9, 8 ( ) and G 5, 6 ( ). Prove that DEFG is a rhombus.

Answer 1

a rhombus is a parallelogram whose four sides are equal length.
first, prove that the opposite sides are parallel by finding the slope of each side:
Slope of DE: (4-2)/(7-3)=1/2
Slope of EF: 2
slope of FG: 1/2
slope of GD: 2
DE is parallel to FG because they have the same slope
EF is parallel to GD
Therefore, DEFG is a parallelogram.

Next, prove all the sides are the same length by finding the distance between the points:
use the distance formula to find distance between two points:
DE=√(4+16)=√20=2√5
EF=2√5
FG=2√5
FD=2√5
The opposite sides are parallel, and all the sides are the same length, so quadrilateral DEFG is a rhombus