Question

2) Quadrilateral ABCD has vertices A(-6,3), B(-3,6), C(9,6), and D(-5,-8). 1 V Using coordinate geometry, prove that (1) ABCD is a trapezoid (2) not an isosceles trapezoid

Answer 1

To prove that quadrilateral ABCD is a trapezoid, we need to show that it has one pair of parallel sides. To prove that ABCD is not an isosceles trapezoid, we need to show that its non-parallel sides are not congruent.

Step-by-step explanation:

To prove that quadrilateral ABCD is a trapezoid, we need to show that it has one pair of parallel sides.

1. Using the slope formula, we find the slopes of opposite sides: AB and CD have slopes of 3/3 = 1, while BC and AD have slopes of 0.
2. Since the slopes of AB and CD are equal (1 = 1), and the slopes of BC and AD are equal (0 = 0), we can conclude that AB || CD and BC || AD, making ABCD a trapezoid.

To prove that ABCD is not an isosceles trapezoid, we need to show that its non-parallel sides are not congruent.

1. Using the distance formula, we find the lengths of BC and AD: BC = √(9-(-3))^2 + (6-6)^2 = √12^2 + 0 = 12. AD = √(-5-(-6))^2 + (-8-3)^2 = √1^2 + (-11)^2 = √1 + 121 = √122.
2. Since BC (12) is not equal to AD (√122), we can conclude that ABCD is not an isosceles trapezoid.

Answer 2

To prove that the points ABCD form a trapezoid, we must first plot the points on the cartesian (x,y) plane. The figure below is such a plot:

Now, one crucial property of any trapezoid is that it must have two opposites sides that are parallel to each other. The sketched plot above already gives us an idea of the sides that could be exactly parallel to each other, but the question asks that we carry out the proof using 'coordinate geometry'. Therefore, we have to apply our knowledge of coordinate geometry to carry out the proof.

According to coordinate geometry, two straight lines are said to be parallel to each other only if they have the sam