Question

select the correct answer.
a quadrilateral has vertices ( a(3,5), b(2,0), c(7,0) ), and ( d(8,5) ). which statement about the quadrilateral is true?
a. ( a b c d ) is a parallelogram with non - perpendicular adjacent sides.
b. ( a b c d ) is a trapezoid with only one pair of parallel sides.
c. ( a b c d ) is a rectangle with non - congruent adjacent sides.
d. ( a b c d ) is a rhombus with non - perpendicular adjacent sides.

Explanation:

Step1: Calculate the slopes of the sides

The slope formula is \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
For side \(AB\) with \(A(3,5)\) and \(B(2,0)\):
\(m_{AB}=\frac{0 - 5}{2 - 3}=\frac{- 5}{-1}=5\)
For side \(BC\) with \(B(2,0)\) and \(C(7,0)\):
\(m_{BC}=\frac{0 - 0}{7 - 2}=0\)
For side \(CD\) with \(C(7,0)\) and \(D(8,5)\):
\(m_{CD}=\frac{5 - 0}{8 - 7}=5\)
For side \(DA\) with \(D(8,5)\) and \(A(3,5)\):
\(m_{DA}=\frac{5 - 5}{3 - 8}=0\)

Step2: Check the properties of the quadrilateral

Since \(m_{AB}=m_{CD}=5\) and \(m_{BC}=m_{DA}=0\), \(AB\parallel CD\) and \(BC\parallel DA\). So \(ABCD\) is a parallelogram.
The product of the slopes of \(AB\) and \(BC\) is \(5\times0 = 0
eq - 1\), so adjacent sides are not perpendicular.
Also, \(AB=\sqrt{(3 - 2)^2+(5 - 0)^2}=\sqrt{1 + 25}=\sqrt{26}\), \(BC=\sqrt{(7 - 2)^2+(0 - 0)^2}=5\). Since \(AB
eq BC\), it is not a rhombus or a rectangle.
A trapezoid has only one pair of parallel sides, but here we have two pairs of parallel sides.

Answer:

A. \(ABCD\) is a parallelogram with non - perpendicular adjacent sides.