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 [tex](x_{1},y_{1}), (x_{2},y_{2})[/tex] to be parallel to [tex](x_{3},y_{3}), (x_{4},y_{4})[/tex] is matching slopes;
[tex]\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})[/tex]---->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],
[tex](5-3) \cdot (-2-3) = (-2-1) \cdot (-1-(-6))2 \cdot -5 = -3 \cdot 5[/tex]
-10 ≠ -15
Similarly,
check AD || BC
A(−6, 3) , D(−2, −2) , B(−1, 5) and C(3, 1)
Substitute in [1], we have
[tex](-2-3) \cdot (3-(-1)) = (1-5) \cdot (-2-(-6))-5 \cdot 4 = -4 \cdot 4[/tex]
-20 ≠ -16.
Both pairs of sides are not parallel,
therefore, Quadrilateral ABCD is not a rhombus because there are no pairs of parallel sides.
