First we will verify that it is a trapezoid, and then we will determine whether the figure is an isosceles trapezoid.
Is It a Trapezoid?
To determine if our quadrilateral is a trapezoid, we have to check if our quadrilateral has exactly one pair of parallel sides. To do this, let's find the slope of each side using the Slope Formula.
Side
Slope Formula
Simplified
Slope of SR: ( -1,-6), ( -2,1)
1-( -6)/-2-( -1)
- 7
Slope of RQ: ( -2,1), ( 2, 5)
5- 1/2-( -2)
1
Slope of QT: ( 2,5), ( 9,4)
4- 5/9- 2
- 1/7
Slope of TS: ( 9,4), ( -1,-6)
-6- 4/-1- 9
1
We can see that the slopes of SR and QT are not equal, so these sides are not parallel. The slopes of RQ and TS are equal, so these sides are parallel. Since our quadrilateral has exactly one pair of parallel sides, it is a trapezoid.
Is It an Isosceles Trapezoid?
A trapezoid is isosceles if its non-parallel sides are congruent. Therefore, we want to check whether the lengths of SR and QT are equal. To do this, we will use the Distance Formula.
Side
Distance Formula
Simplified
Length of SR: ( -1,-6), ( -2,1)
sqrt(( - 2-( -1))^2+( 1-( -6))^2)
sqrt(50)
Length of QT: ( 2,5), ( 9,4)
sqrt(( 9- 2)^2+( 4- 5)^2)
sqrt(50)
Since the lengths are equal, SR and QT are congruent. Therefore, QRST is an isosceles trapezoid.
