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A quadrilateral has vertices A(3, 5), B(2, 0), C(7, 0), and D(8, 5). Which statement about the quadrilateral is true?

AABCD is a parallelogram with non-perpendicular adjacent sides.
BABCD is a trapezoid with only one pair of parallel sides.
CABCD is a rectangle with non-congruent adjacent sides.

D)ABCD is a rhombus with non-perpendicular adjacent sides.

Verified Answer :

W0lf93

In the question "A quadrilateral has vertices A(3, 5), B(2, 0), C(7, 0), and D(8, 5). Which statement about the quadrilateral is true?"
Line BC is parallel to line AD because their slopes is equal i.e. (0 - 0) / (7 - 2) = (5 - 5) / (8 - 3) which gives 0 / 5 = 0 / 5 giving that 0 = 0.
We check whether line AB is parallel to line CD. Slope of line AB is given by (0 - 5) / (2 - 3) = -5 / -1 = 5. Slope of line CD is given by (5 - 0) / (8 - 7) = 5 / 1 = 5
We have been able to prove that the opposite sides of the quadrilateral are parallel which means that the quadrilateral is not a trapezoid.
Next we check whether the length of the sides are equal.
Length of line AB is given by sqrt[(0 - 5)^2 + (2 - 3)^2] = sqrt[(-5)^2 + (-1)^2] = sqrt(25 + 1) = sqrt(26)
Length of line BC is given by sqrt[(0 - 0)^2 + (7 - 2)^2] = sqrt[0^2 + 5^2] = sqrt(25) = 5
Length of line CD is given by sqrt[(5 - 0)^2 + (8 - 7)^2] = sqrt[5^2 + 1^2] = sqrt(25 + 1) = sqrt(26)
Length of line DA is given by sqrt[(5 - 5)^2 + (8 - 3)^2] = sqrt[0^2 + 5^2] = sqrt(25) = 5
Thus, the length of the sides of the quadrilateral are not equal but opposite sides are equal which means that the quadrilateral is not a rhombus.
Finally, we check whether adjacent lines are perpendicular. Recall the for perpendicular lines, the product of their slopes is equal to -1.
Slope of line AB = 5 while slope of line BC = 0. The product of their slopes = 5 x 0 = 0 which is not -1, thus the adjacent sides of the quadrilateral are not perpendicular which means that the quadrilateral is not a rectangle.
Therefore, ABCD is a parallelogram with non-perpendicular adjacent sides. (option A).

frika

A quadrilateral has vertices A(3, 5), B(2, 0), C(7, 0), and D(8, 5). Find:

\(\overrightarrow{AB}=(x_B-x_A,y_B-y_A)=(2-3,0-5)=(-1,-5);\\ \\ \overrightarrow{BC}=(x_C-x_B,y_C-y_B)=(7-2,0-0)=(5,0);\\ \\ \overrightarrow{CD}=(x_D-x_C,y_D-y_C)=(8-7,5-0)=(1,5);\\ \\ \overrightarrow{DA}=(x_A-x_D,y_A-y_D)=(3-8,5-5)=(-5,0).\)

You can see that vectors \(\overrightarrow{AB}||\overrightarrow{CD}\) and  \(\overrightarrow{BC}||\overrightarrow{DA}.\) This means that opposite sides are parallel and option B is false.

Find the lengths of all sides:

\(|\overrightarrow{AB}|=\sqrt{(-1)^2+(-5)^2}=\sqrt{26},\\ \\ |\overrightarrow{BC}|=\sqrt{(5)^2+(0)^2}=\sqrt{25}=5,\\ \\ |\overrightarrow{CD}|=\sqrt{(1)^2+(5)^2}=\sqrt{26},\\ \\ |\overrightarrow{DA}|=\sqrt{(-5)^2+(0)^2}=\sqrt{25}=5.\)

As you can see the lengths of opposite sides are equal, but all lengths are not equal. Therefore, the last option D is false.

Now check whether angles A and B are perpendicular:

\(\overrightarrow{AB}\cdot \overrightarrow{BC}=-1\cdot 5+(-5)\cdot 0=-5 eq 0;\\ \\ \overrightarrow{AB}\cdot \overrightarrow{DA}=-1\cdot (-5)+(-5)\cdot 0=5 eq 0.\)

The dot products are not equal to zero, then angles A and B are nor right. This means that option C is false and option A is correct (ABCD is a parallelogram with non-perpendicular adjacent sides).

Answer: correct choice is A.

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