Question

Determine whether the quadrilateral is a parallelogram using the specified method. 13. D(-8, 1), E(-3, 6), F(7, 4), G(2, -1) (Distance Formula) 14. L(-1, 6), M(5, 9), N(0, 2), P(-8, -2) (Slope Formula) 15. B(-2, -9), C(0, -5), D(6, -3), T(4, -7) (Distance and Slope Formulas)

a) (13) Yes, (14) No, (15) No

b) (13) No, (14) Yes, (15) Yes

c) (13) Yes, (14) Yes, (15) No

d) (13) No, (14) No, (15) Yes

Answer 1

To determine whether the quadrilateral is a parallelogram using the Distance Formula, calculate the distances between opposite sides of the quadrilateral and compare them. If the distances are equal, the quadrilateral is a parallelogram.

Step-by-step explanation:

To determine whether the quadrilateral is a parallelogram using the Distance Formula, we need to calculate the distances between opposite sides of the quadrilateral. Let's calculate the distances for each given set of points:

1. For points D and E: Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2) = sqrt((-3 - (-8))^2 + (6 - 1)^2) = sqrt(25 + 25) = sqrt(50)
2. For points E and F: Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2) = sqrt((7 - (-3))^2 + (4 - 6)^2) = sqrt(100 + 4) = sqrt(104)
3. For points F and G: Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2) = sqrt((2 - 7)^2 + (-1 - 4)^2) = sqrt(25 + 25) = sqrt(50)
4. For points G and D: Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2) = sqrt((-8 - 2)^2 + (1 - (-1))^2) = sqrt(100 + 4) = sqrt(104)

If the opposite sides of the quadrilateral have the same lengths, then it is a parallelogram. Comparing the distances, we can see that the distance between D and E is sqrt(50) and the distance between G and F is sqrt(50), which are equal. Additionally, the distance between E and F is sqrt(104) and the distance between G and D is sqrt(104), which are also equal. Therefore, the quadrilateral is a parallelogram.