Question

The coordinates of the vertices of quadrilateral ABCD are A(−6, 3) , B(−1, 5) , C(3, 1) , and D(−2, −2) . Which statement correctly describes whether quadrilateral ABCD is a rhombus?

Answer 1

C.Quadrilateral ABCD is not a rhombus because there are no pairs of parallel sides.

Complete question:

A. Quadrilateral ABCD is not a rhombus because opposite sides are parallel but the four sides do not all have the same length.

B. Quadrilateral ABCD is a rhombus because opposite sides are parallel and all four sides have the same length.

C. Quadrilateral ABCD is not a rhombus because there are no pairs of parallel sides.

D. Quadrilateral ABCD is not a rhombus because there is only one pair of opposite sides that are parallel.

Explanation:

Rhombus states that a parallelogram with four equal sides and sometimes one with no right angle.

Given: The coordinate of the vertices of quadrilateral ABCD are A(−6, 3) , B(−1, 5) , C(3, 1) , and D(−2, −2) .

The condition for the segment
`(x_(1),y_(1)), (x_(2),y_(2))` to be parallel to
`(x_(3),y_(3)), (x_(4),y_(4))` is matching slopes;

`(y_(2)-y_(1))/(x_(2)-x_(1))= (y_(4)-y_(3))/(x_(4)-x_(3)) \\(y_(2)-y_(1)) \cdot (x_(4)-x_(3)) =(y_(4)-y_(3)) \cdot (x_(2)-x_(1))`---->1

So, we have to check that AB || CD and AD || BC

First check AB || CD

A(−6, 3) , B(−1, 5) , C(3, 1) , and D(−2, −2)

substitute in [1],

`(5-3) \cdot (-2-3) = (-2-1) \cdot (-1-(-6))2 \cdot -5 = -3 \cdot 5`

-10 ≠ -15

Similarly,

check AD || BC

A(−6, 3) , D(−2, −2) , B(−1, 5) and C(3, 1)

Substitute in [1], we have

`(-2-3) \cdot (3-(-1)) = (1-5) \cdot (-2-(-6))-5 \cdot 4 = -4 \cdot 4`

-20 ≠ -16.

Both pairs of sides are not parallel,

therefore, Quadrilateral ABCD is not a rhombus because there are no pairs of parallel sides.