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 the slope of each side

The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$.
Slope of $AB$ with $A(4,5)$ and $B(2,4)$: $m_{AB}=\frac{4 - 5}{2 - 4}=\frac{-1}{-2}=\frac{1}{2}$.
Slope of $BC$ with $B(2,4)$ and $C(4,3)$: $m_{BC}=\frac{3 - 4}{4 - 2}=-\frac{1}{2}$.
Slope of $CD$ with $C(4,3)$ and $D(6,4)$: $m_{CD}=\frac{4 - 3}{6 - 4}=\frac{1}{2}$.
Slope of $DA$ with $D(6,4)$ and $A(4,5)$: $m_{DA}=\frac{5 - 4}{4 - 6}=-\frac{1}{2}$.
Since $m_{AB}=m_{CD}$ and $m_{BC}=m_{DA}$, opposite - sides are parallel.

Step2: Calculate the length of adjacent sides

The distance formula is $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.
Length of $AB$: $d_{AB}=\sqrt{(2 - 4)^2+(4 - 5)^2}=\sqrt{(-2)^2+(-1)^2}=\sqrt{4 + 1}=\sqrt{5}$.
Length of $BC$: $d_{BC}=\sqrt{(4 - 2)^2+(3 - 4)^2}=\sqrt{2^2+(-1)^2}=\sqrt{4 + 1}=\sqrt{5}$.
Adjacent sides are congruent.

Step3: Check the product of slopes of adjacent sides

$m_{AB}\times m_{BC}=\frac{1}{2}\times(-\frac{1}{2})=-\frac{1}{4}
eq - 1$, so adjacent sides are not perpendicular.

Answer:

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