Question

Which statement BEST describes a parallelogram with coordinates A (-1, 0), B (-4, 5), C (1, 8) and D (4, 3)? A. The diagonals are congruent which means the quadrilateral is a rectangle. B. The diagonals are perpendicular which means the quadrilateral is a rhombus. C. The diagonals are both congruent and perpendicular which means the quadrilateral is a square. D. The diagonals are neither congruent nor perpendicular, which means the quadrilateral is a parallelogram only.

Answer 1

Answer:

D.

Step-by-step explanation:

The representation of this problem is shown below. To find the answer, we need to use the distance formula:

$The \ \mathbf{distance} \ d \ between \ the \ \mathbf{points} \ (x_{1},y_{1}) \ and \ (x_{2},y_{2}) \ in \ the \ plane \ is:\\ \\ d=\sqrt{(x_{2}-x_{1})^2+(y_{2}-y_{1})^2}$

• The first diagonal is formed by the points A and C
• The second diagonal is formed by the points B and D

So, for the first diagonal:

$A(x_{1},y_{1})=A(-1,10) \\ \\ C(x_{2},y_{2})=C(1,8) \\ \\ \\ d_{1}=\sqrt{(x_{2}-x_{1})^2+(y_{2}-y_{1})^2} \\ \\ d_{1}=\sqrt{[1-(-1)]^2+(8-10)^2}=2\sqrt{2}$

For the second diagonal:

$B(x_{1},y_{1})=B(-4,5) \\ \\ D(x_{2},y_{2})=D(4,3) \\ \\ \\ d_{2}=\sqrt{(x_{2}-x_{1})^2+(y_{2}-y_{1})^2} \\ \\ d_{2}=\sqrt{[4-(-4)]^2+(3-5)^2}=2\sqrt{17}$

So the diagonals aren't congruent. Are they perpendicular?

$\vec{AC}=(1,8)-(-1,10)=(2,-2) \\ \\ \vec{BD}=(4,3)-(-4,5)=(8,-2)$

These two vectors will be perpendicular (hence the diagonals will be perpendicular) if and only if the dot product equals zero, so:

$(2,-2).(8,-2)=2(8)+(-2)(-2)=20\neq 0$

Thus, the diagonals aren't perpendicular. In conclusion:

D. The diagonals are neither congruent nor perpendicular, which means the quadrilateral is a parallelogram only.