Answer:
Option C is correct.
Step-by-step explanation:
Given a quadrilateral has vertices A(4, 9), B(2, 5), C(8, 2), and D(10, 6). we have to check the statements which are true.
Using distance formula,
\(Distance=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\)
\(AB=\sqrt{(2-4)^2+(5-9)^2}=\sqrt{4+16}=\sqrt{20}units\)
\(BC=\sqrt{(8-2)^2+(2-5)^2}=\sqrt{36+9}=\sqrt{45}units\)
\(CD=\sqrt{(10-8)^2+(6-2)^2}=\sqrt{4+16}=\sqrt{20}units\)
\(AD=\sqrt{(10-4)^2+(6-9)^2}=\sqrt{36+9}=\sqrt{45}units\)
Opposite sides of quadrilateral are equal.
Now, we have to find the slope of sides to find the angle between the sides of quadrilateral.
\(\text{slope of AB=}\frac{y_2-y_1}{x_2-x_1}=\frac{5-9}{2-4}=2\)
\(\text{slope of BC=}\frac{y_2-y_1}{x_2-x_1}=\frac{2-5}{8-2}=\frac{-1}{2}\)
\(\text{slope of CD=}\frac{y_2-y_1}{x_2-x_1}=\frac{6-2}{10-8}=2\)
\(\text{slope of AD=}\frac{y_2-y_1}{x_2-x_1}=\frac{6-9}{10-4}=\frac{-1}{2}\)
Adjacent sides has the slopes negative reciprocals.
⇒ Adjacent sides of the quadrilateral are perpendicular.
hence, ABCD is a rectangle with non-congruent adjacent sides.
Option C is correct.
