Question

Determine whether quadrilateral ABCD with vertices A(4,-5), B(-3,0), C(0, 2), and D(5, 1) is a trapezoid.

Step 1: Find the slope of AB. The slope of AB is
a) ( 5/7 )
b) ( -5/7 )
c) ( -7/5 )
d) ( 7/5 )

Step 2: Find the slope of DC. The slope of DC is
a) ( 3/5 )
b) ( -3/5 )
c) ( -5/3 )
d) ( 5/3 )

Step 3: Find the slope of BC. The slope of BC is
a) ( 2/3 )
b) ( -3/2 )
c) ( -3/4 )
d) ( 3/4 )

Step 4: Find the slope of AD. The slope AD is
a) ( 1/9 )
b) ( -1/9 )
c) ( -9/1 )
d) ( 9/1 )

The quadrilateral is a trapezoid because
a) Opposite sides are parallel.
b) Opposite sides have equal lengths.
c) Diagonals bisect each other.
d) All angles are right angles.

Answer 1

After calculating the slopes of the sides of quadrilateral ABCD: AB (-5/7), DC (-1/5), BC (2/3), and AD (6/1), it's clear that there are no pairs of opposite sides with identical slopes, so ABCD is not a trapezoid.

Step-by-step explanation:

To determine whether quadrilateral ABCD is a trapezoid, we need to calculate the slopes of its sides. A trapezoid is a quadrilateral with at least one pair of opposite sides that are parallel.

1. Find the slope of AB: The slope is calculated using the coordinates of points A and B (4,-5) and (-3,0), resulting in (change in y) / (change in x) = (0 - (-5)) / (-3 - 4) which simplifies to 5 / -7 or -5/7.
2. Find the slope of DC: Using points D and C (5, 1) and (0, 2), we get the slope as (1 - 2) / (5 - 0) which is -1 / 5 or -1/5. Since this is not the negative reciprocal of AB's slope, AB and CD are not perpendicular. Moreover, the slopes are not the same, so AB and CD are not parallel.
3. Find the slope of BC: Using the formula for slope with points B and C (-3,0) and (0, 2), we get (2 - 0) / (0 - (-3)) which is 2 / 3 or 2/3.
4. Find the slope of AD: We use points A and D (4,-5) and (5, 1) to find the slope as (1 - (-5)) / (5 - 4) which equals 6 / 1 or 6/1.

In order for ABCD to be a trapezoid, we need to find at least one pair of opposite sides that are parallel (same slope). From our calculations, no pair of opposite sides have the same slope, so ABCD is not a trapezoid, and none of the options (a) through (d) apply.