Quadrilateral with two pairs of consecutive sides congruent and no opposite sides congruent
Now, let's find the slopes of the sides using the Slope Formula.
Side
Slope Formula
Simplified
Slope of QU: ( 4,5), ( 12,14)
14- 5/12- 4
9/8
Slope of UA: ( 12,14), ( 20, 5)
5- 14/20- 12
- 9/8
Slope of AD: ( 20, 5), ( 12, -4)
-4- 5/12- 20
9/8
Slope of DQ: ( 12, -4), ( 4,5)
5-( -4)/4- 12
- 9/8
We can tell that the slopes of the opposite sides of our quadrilateral are equal. Therefore, both pairs of opposite sides are parallel and the quadrilateral is a parallelogram. Let's multiply the slopes of two consecutive sides
9/8 ( - 9/8 ) = - 81/64
Since the product is - 8164≠- 1 the sides are not opposite reciprocals. Therefore, the quadrilateral is either a parallelogram or a rhombus. To check, we can find the lengths of its sides using the Distance Formula.
Side
Distance Formula
Simplified
Length of QU: ( 4,5), ( 12,14)
sqrt(( 12- 4)^2+( 14- 5)^2)
sqrt(145)
Length of UA: ( 12,14), ( 20, 5)
sqrt(( 20- 12)^2+( 5- 14)^2)
sqrt(145)
Length of AD: ( 20, 5), ( 12, -4)
sqrt(( 12- 20)^2+( -4- 5)^2)
sqrt(145)
Length of DQ: ( 12, -4), ( 4,5)
sqrt(( 4- 12)^2+( 5-( -4))^2)
sqrt(145)
Our parallelogram has four congruent sides. Therefore, the most precise name for this quadrilateral is a rhombus.
