Question

Determining Geometric Properties of a Figure Using the Slopes of Lines
A quadrilateral has vertices E(–4, 2), F(4, 7), G(8, 1), and H(0, –4). Which statements are true? Check all that apply.

The slope of EH is .
The slopes of EF and GH are both .
FG is perpendicular to GH.
Quadrilateral EFGH is a parallelogram because both pairs of opposite sides are parallel.
Quadrilateral EFGH is a rectangle because all angles are right angles.

Answer 1

Compute slopes:

- EH: slope = (−4 − 2)/(0 − (−4)) = −6/4 = −3/2. ✓
- EF: slope = (7 − 2)/(4 − (−4)) = 5/8.
- GH: slope = (−4 − 1)/(0 − 8) = −5/−8 = 5/8. EF and GH both have slope 5/8. ✓
- FG: slope = (1 − 7)/(8 − 4) = −6/4 = −3/2.

Check perpendicular: product of slopes of FG (−3/2) and GH (5/8) = −15/16 ≠ −1, so FG is not perpendicular to GH. ✗

Opposite sides: EF ∥ GH and FG ∥ EH, so EFGH is a parallelogram. ✓

All angles right? No (adjacent slopes multiply to −15/16, not −1), so not a rectangle. ✗

True statements:
- The slope of EH is −3/2.
- The slopes of EF and GH are both 5/8.
- Quadrilateral EFGH is a parallelogram because both pairs of opposite sides are parallel.

False statements:
- FG is perpendicular to GH.
- Quadrilateral EFGH is a rectangle.