Question

A quadrilateral has vertices A(4, 9), B(2, 5), C(8, 2), and D(10, 6). Which statement about the quadrilateral is true?
(A) ABCD is a parallelogram with non-perpendicular 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.

Answer 1

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.

Answer 2

The true statement about the given quadrilateral is (C) ABCD is a rectangle with noncongruent adjacent sides.

Calculations and Parameters

Here , A(4, 9), B(2, 5), C(8, 2), and D(10, 6),

If we use the distance formula,

AB= $\sqrt{16 + 4}$= $\sqrt{20}$ units

BC= $\sqrt{45}$

CD= $\sqrt{20}$

DA= $\sqrt{45}$

Slope of AB= 2

Slope of BC= -0.5

Slope AB * Slope BC= -1

This shows that they are perpendicular to each other.

Slope CD= 2

Slope DA= -0.5

Slope CD * DA= -1

This shows that they are perpendicular to each other

The quadrilateral ABCD is a rectangle with noncongruent adjacent sides.

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