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 it has exactly one pair of parallel sides. To do this, let's find the slope of each side using the Slope Formula.
Side
m=y_2-y_1/x_2-x_1
Simplified
Slope of QR: ( - 12,1), ( - 9,4)
4- 1/- 9-( - 12)
1
Slope of RS: ( - 9,4), ( - 4,3)
3- 4/- 4-( - 9)
- 1/5
Slope of ST: ( - 4,3), ( - 11,- 4)
- 4- 3/- 11-( - 4)
1
Slope of TQ: ( - 11,- 4), ( - 12,1)
1-( - 4)/- 12-( - 11)
- 5
We can see that the slopes of RS and TQ are not equal, so these sides are not parallel. The slopes of QR and ST 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 RS and TQ are equal. To do this, we will use the Distance Formula.
Side
d=sqrt((x_2-x_1 )^2+(y_2-y_1 )^2)
Simplified
Length of RS: ( - 9,4), ( - 4,3)
sqrt(( - 4-( - 9))^2+( 3- 4)^2)
sqrt(26)
Length of TQ: ( - 11,- 4), ( - 12,1)
sqrt(( - 12-( - 11))^2+( 1-( - 4))^2)
sqrt(26)
Since the lengths are equal, RS and TQ are congruent. Therefore, QRST is an isosceles trapezoid.
