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

D.

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)=√([1-(-1)]^2+(8-10)^2)=2√(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)=√([4-(-4)]^2+(3-5)^2)=2√(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\\eq 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.