Question

question 3 of 5 select the correct answer. quadrilateral abcd has vertices of a(10,2), b(2, - 4), c(-4,4), and d(4,10). which statement is true about the quadrilateral? abcd is a trapezoid with only one pair of parallel sides. abcd is a parallelogram with non - perpendicular adjacent sides. abcd is a rectangle with non - congruent adjacent sides. abcd is a square with 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 $A(10,2)$ and $B(2,-4)$:
$m_{AB}=\frac{-4 - 2}{2 - 10}=\frac{-6}{-8}=\frac{3}{4}$
For side BC with $B(2,-4)$ and $C(-4,4)$:
$m_{BC}=\frac{4+4}{-4 - 2}=\frac{8}{-6}=-\frac{4}{3}$
For side CD with $C(-4,4)$ and $D(4,10)$:
$m_{CD}=\frac{10 - 4}{4+4}=\frac{6}{8}=\frac{3}{4}$
For side DA with $D(4,10)$ and $A(10,2)$:
$m_{DA}=\frac{2 - 10}{10 - 4}=\frac{-8}{6}=-\frac{4}{3}$

Step2: Analyze parallel and perpendicular sides

Since $m_{AB}=m_{CD}=\frac{3}{4}$ and $m_{BC}=m_{DA}=-\frac{4}{3}$, opposite - sides are parallel.
Also, $m_{AB}\times m_{BC}=\frac{3}{4}\times(-\frac{4}{3})=- 1$, adjacent sides are perpendicular.

Step3: Calculate the lengths of the sides

The distance formula is $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.
For side AB:
$d_{AB}=\sqrt{(2 - 10)^2+(-4 - 2)^2}=\sqrt{(-8)^2+(-6)^2}=\sqrt{64 + 36}=\sqrt{100}=10$
For side BC:
$d_{BC}=\sqrt{(-4 - 2)^2+(4 + 4)^2}=\sqrt{(-6)^2+8^2}=\sqrt{36+64}=\sqrt{100}=10$
Since all sides are equal (AB = BC = CD = DA = 10) and adjacent sides are perpendicular, the quadrilateral is a square.

Answer:

ABCD is a square with perpendicular adjacent sides.