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?

a. ( abcd ) is a parallelogram with non - perpendicular adjacent sides.

b. ( abcd ) is a trapezoid with only one pair of parallel sides.

c. ( abcd ) is a rectangle with non - congruent adjacent sides.

d. ( abcd ) 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 parallelogram

Since \(m_{AB}=m_{CD} = 5\) and \(m_{BC}=m_{DA}=0\), opposite sides are parallel.
Now check if adjacent sides are perpendicular. The product of slopes of \(AB\) and \(BC\) is \(5\times0 = 0
eq - 1\) (for perpendicular lines \(m_1\times m_2=-1\)).

Answer:

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