Let's plot the given points on a coordinate plane and connect them to form the quadrilateral.
Then we will determine whether or not the quadrilateral is a parallelogram. Here we are asked to use the Distance and Slope Formulas.
Let's recall that if one pair of opposite sides is both parallel and congruent, then the quadrilateral is a parallelogram. Therefore, we will consider SR and TQ and check whether they are parallel and congruent.
Parallel
First, let's check if the sides are parallel. To do so, we will compare their slopes.
Side
Slope Formula
Simplify
Slope of SR: ( -3,6), ( 4,3)
3- 6/4-( -3)
- 3/7
Slope of TQ: (-5, -1), (2, -4)
-4-(-1)/2-(-5)
- 3/7
The slopes of SR and TQ are equal, so these sides are parallel.
Congruent
Now, let's check if the sides are congruent by comparing their lengths.
Side
Distance Formula
Simplify
Length of SR: ( -3,6), ( 4,3)
sqrt(( 4-( -3))^2+( 3- 6)^2)
sqrt(58)
Length of TQ: (-5, -1), (2, -4)
sqrt((2-(-5))^2+(-4-(-1))^2)
sqrt(58)
The lengths of the sides are equal, so SR and TQ are congruent.
Conclusion
The opposite sides SR and TQ are parallel and congruent, so the given quadrilateral is a parallelogram.
