Question

Determine whether a figure with the given vertices is a rectangle using the Distance Formula.

Answer 1

Answer:

Yes; Opposite sides are congruent, and diagonals are congruent.

Step-by-step explanation:

we have

$A(4, -7), B(4, -2), C(0, -2), D(0, -7)$

we know that

the formula to calculate the distance between two points is equal to

$d=\sqrt{(y2-y1)^{2}+(x2-x1)^{2}}$

step 1

Find the length of the sides

Find the distance AB

substitute the values

$d=\sqrt{(-2+7)^{2}+(4-4)^{2}}$

$d=\sqrt{(5)^{2}+(0)^{2}}$

$AB=5\ units$

Find the distance BC

substitute the values

$d=\sqrt{(-2+2)^{2}+(0-4)^{2}}$

$d=\sqrt{(0)^{2}+(-4)^{2}}$

$BC=4\ units$

Find the distance CD

substitute the values

$d=\sqrt{(-7+2)^{2}+(0-0)^{2}}$

$d=\sqrt{(-5)^{2}+(0)^{2}}$

$CD=5\ units$

Find the distance AD

substitute the values

$d=\sqrt{(-7+7)^{2}+(0-4)^{2}}$

$d=\sqrt{(0)^{2}+(-4)^{2}}$

$AD=4\ units$

Compare the length sides

AB=CD

BC=AD

therefore

Opposite sides are congruent

step 2

Find the length of the diagonals

Find the distance AC

substitute the values

$d=\sqrt{(-2+7)^{2}+(0-4)^{2}}$

$d=\sqrt{(5)^{2}+(-4)^{2}}$

$AC=\sqrt{41}\ units$

Find the distance BD

substitute the values

$d=\sqrt{(-7+2)^{2}+(0-4)^{2}}$

$d=\sqrt{(-5)^{2}+(-4)^{2}}$

$BD=\sqrt{41}\ units$

Compare the length of the diagonals

AC=BD

therefore

Diagonals are congruent

The figure is a rectangle, because Opposite sides are congruent, and diagonals are congruent

Answer 2

Answer:

The correct option is 2.

Step-by-step explanation:

Given information: A(4, –7), B(4, –2), C(0, –2), D(0, –7).

Distance formula:

$D=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

Using distance formula we get

Length of sides are

$AB=\sqrt{\left(4-4\right)^2+\left(-2-\left(-7\right)\right)^2}$

$AB=\sqrt{(0)^2+(-2+7)^2}$

$AB=\sqrt{5^2}=5$

Similarly,

$BC=\sqrt{\left(0-4\right)^2+\left(-2-\left(-2\right)\right)^2}=4$

$CD=\sqrt{\left(0-0\right)^2+\left(-7-\left(-2\right)\right)^2}=5$

$AD=\sqrt{\left(0-4\right)^2+\left(-7-\left(-7\right)\right)^2}=4$

Length of diagonals are

$AC=\sqrt{\left(0-4\right)^2+\left(-2-\left(-7\right)\right)^2}=\sqrt{41}$

$BD=\sqrt{\left(0-4\right)^2+\left(-7-\left(-2\right)\right)^2}=\sqrt{41}$

It figure ABCD,

1. AB and CD are opposite sides.

2. BC and AD are opposite sides.

3. AC and BD are diagonals.

From the above calculations it is clear that opposite sides are congruent, and diagonals are congruent. So, the figure ABCD is a rectangle.

Therefore the correct option is 2.