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

Answer:

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.

Step-by-step 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;

$\frac{y_{2}-y_{1}}{x_{2}-x_{1}}= \frac{y_{4}-y_{3}}{x_{4}-x_{3}} \\(y_{2}-y_{1}) \cdot (x_{4}-x_{3}) =(y_{4}-y_{3}) \cdot (x_{2}-x_{1})$---->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.