Question

a quadrilateral has vertices a(4, 5), b(2, 4), c(4, 3), and d(6, 4). which statement about the quadrilateral is true? a. abcd is a parallelogram with noncongruent adjacent sides. b. abcd is a trapezoid with only one pair of parallel sides. c. abcd is a rectangle with noncongruent adjacent sides. d. abcd is a square. e. abcd is a rhombus with non - perpendicular adjacent sides.

Explanation:

Step1: Calculate side lengths

Use distance formula $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$:

• $AB=\sqrt{(2-4)^2+(4-5)^2}=\sqrt{4+1}=\sqrt{5}$
• $BC=\sqrt{(4-2)^2+(3-4)^2}=\sqrt{4+1}=\sqrt{5}$
• $CD=\sqrt{(6-4)^2+(4-3)^2}=\sqrt{4+1}=\sqrt{5}$
• $DA=\sqrt{(4-6)^2+(5-4)^2}=\sqrt{4+1}=\sqrt{5}$

Step2: Calculate side slopes

Use slope formula $m=\frac{y_2-y_1}{x_2-x_1}$:

• Slope of $AB$: $m_{AB}=\frac{4-5}{2-4}=\frac{-1}{-2}=\frac{1}{2}$
• Slope of $BC$: $m_{BC}=\frac{3-4}{4-2}=\frac{-1}{2}=-\frac{1}{2}$
• Slope of $CD$: $m_{CD}=\frac{4-3}{6-4}=\frac{1}{2}$
• Slope of $DA$: $m_{DA}=\frac{5-4}{4-6}=\frac{1}{-2}=-\frac{1}{2}$

Step3: Check parallel/perpendicular sides

• Parallel: $m_{AB}=m_{CD}$, $m_{BC}=m_{DA}$ (2 pairs of parallels)
• Perpendicular: Product of adjacent slopes $\frac{1}{2} \times -\frac{1}{2}=-\frac{1}{4}

eq -1$ (not perpendicular)

Step4: Match to quadrilateral type

All sides congruent, opposite sides parallel, adjacent sides not perpendicular.

Answer:

E. ABCD is a rhombus with non-perpendicular adjacent sides.