Answer: (C) ABCD is a rectangle with noncongruent adjacent sides.
Step-by-step explanation:
Here , A(4, 9), B(2, 5), C(8, 2), and D(10, 6),
Thus, by the distance formula,
\(AB =\sqrt{(5-9)^2+(2-4)^2}=\sqrt{16+4}=\sqrt{20}\text{ unit}\)
Similarly,
\(BC = \sqrt{45}\text{ unit}\),
\(CD=\sqrt{20}\text{ unit}\),
\(DA=\sqrt{45}\text{ unit}\)
Hence, for the quadrilateral,
AB = CD and BC = DA ---------(1)
Now, the slope of AB
\(= \frac{5-9}{2-4}=\frac{-4}{-2}=2\)
the slope of BC
\(=\frac{2-5}{8-2}=\frac{-3}{6}=-\frac{1}{2}\)
Slope of AB × Slope of BC = 2 × -1/2 = -1
⇒ Sides AB and BC are perpendicular to each other ----- (2)
Now, the slope of CD
\(= \frac{6-2}{10-8}=\frac{4}{2}=2\)
the slope of DA
\(=\frac{9-6}{4-10}=\frac{3}{-6}=-\frac{1}{2}\)
Slope of CD × Slope of DA = 2 × -1/2 = -1
⇒ Sides CD and DA are perpendicular to each other ----- (3)
Hence, from equations (1), (2) and (3),
The quadrilateral ABCD is a rectangle with non congruent adjacent sides.
