Question

Quadrilateral A B C D has vertices A(-3,4), B(5,2), C(6,-4), and D(-6,-1).

Show that A B C D is a trapezoid.

Answer 1

The quadrilateral ABCD formed by the vertices A(-3,4), B(5,2), C(6,-4), and D(-6,-1) is a parallelogram.

Step-by-step explanation:

To determine if ABCD is a parallelogram, we can use the slope formula to analyze the slopes of opposite sides. The slope of side AB is (2 - 4) / (5 - (-3)) = -2/8 = -1/4, and the slope of side CD is (-1 - (-4)) / (-6 - 6) = 3 / (-12) = -1/4. Both slopes are equal, indicating that AB and CD are parallel.

Next, we calculate the slopes of sides BC and AD. The slope of BC is (-4 - 2) / (6 - 5) = -6 / 1 = -6, and the slope of AD is (-1 - 4) / (-6 - (-3)) = -5 / (-3) = 5/3. Since -6 ≠ 5/3, BC and AD are not parallel.

The opposite sides AB and CD are parallel, but BC and AD are not parallel. Therefore, using the definition of a parallelogram where opposite sides are both parallel and equal in length, it confirms that ABCD is a parallelogram.

Answer 2

The quadrilateral ABCD formed by the vertices A(-3,4), B(5,2), C(6,-4), and D(-6,-1) is a parallelogram.

Step-by-step explanation:

To determine if ABCD is a parallelogram, we can use the slope formula to analyze the slopes of opposite sides. The slope of side AB is (2 - 4) / (5 - (-3)) = -2/8 = -1/4, and the slope of side CD is (-1 - (-4)) / (-6 - 6) = 3 / (-12) = -1/4. Both slopes are equal, indicating that AB and CD are parallel.

Next, we calculate the slopes of sides BC and AD. The slope of BC is (-4 - 2) / (6 - 5) = -6 / 1 = -6, and the slope of AD is (-1 - 4) / (-6 - (-3)) = -5 / (-3) = 5/3. Since -6 ≠ 5/3, BC and AD are not parallel.

The opposite sides AB and CD are parallel, but BC and AD are not parallel. Therefore, using the definition of a parallelogram where opposite sides are both parallel and equal in length, it confirms that ABCD is a parallelogram.