To decide who is correct, we need to determine whether the given quadrilateral is a parallelogram and if yes, whether it is also a rectangle. Note that a rectangle is a special case of a parallelogram. Therefore, we will start by investigating if this is a parallelogram.
Is the Quadrilateral a Parallelogram?
For a quadrilateral to be a parallelogram, opposite sides must be parallel. Recall that two sides are parallel if they have the same slope. We can calculate the slopes of opposite sides by using the Slope Formula.
m = y_2-y_1/x_2-x_1
Side
Points
Substitute
Simplify
SP
S( - 2,1) and P( 0,2)
m_(SP)=2- 1/0-( - 2)
m_(SP)=1/2
RQ
R( 1,- 5) and Q( 3,- 4)
m_(RQ)=- 4-( - 5)/3- 1
m_(RQ)=1/2
SR
S( - 2,1) and R( 1,- 5)
m_(SR)=- 5- 1/1-( - 2)
m_(SR)=- 2
PQ
P( 0,2) and Q( 3,- 4)
m_(PQ)=- 4- 2/3- 0
m_(PQ)=- 2
Since opposite sides have the same slope, they are parallel.
This proves that the given quadrilateral is a parallelogram.
Is the Quadrilateral a Rectangle?
For a parallelogram to be a rectangle, adjacent sides must be perpendicular. Slopes of perpendicular sides are opposite reciprocals. This means that the product of their slopes is equal to - 1. Let's verify this.
Sides
Slopes
Product
Simplify
Perpendicular?
SP and SR
m_(SP)= 1/2 and m_(SR)= - 2
1/2( - 2)? =- 1
- 1 = - 1 âś“
Yes
RQ and SR
m_(RQ)= 1/2 and m_(SR)= - 2
1/2( - 2)? =- 1
- 1 = - 1 âś“
Yes
As we can see, SR is perpendicular to both SP and RQ. We have already found that PQ is parallel to SR. As a result, PQ is also perpendicular to both SP and RQ. Therefore, in this quadrilateral, adjacent sides are perpendicular. This means that the parallelogram is also a rectangle.
Therefore, the friend that claims the quadrilateral is a rectangle is correct.
