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Ne4ueva [31]
3 years ago
14

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

Mathematics
2 answers:
Alex Ar [27]3 years ago
8 0

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

<u><em>Find the distance AB</em></u>

substitute the values

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

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

AB=5\ units

<u><em>Find the distance BC</em></u>

substitute the values

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

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

BC=4\ units

<u><em>Find the distance CD</em></u>

substitute the values

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

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

CD=5\ units

<u><em>Find the distance AD</em></u>

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

<u><em>Find the distance AC</em></u>

substitute the values

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

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

AC=\sqrt{41}\ units

<u><em>Find the distance BD</em></u>

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

klasskru [66]3 years ago
8 0

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.

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