On your paper, construct a rectangle on a coordinate plane that satisfies these criteria.

• The sides of the rectangle are either vertical or horizontal.

• The perimeter of the rectangle is 42 units.

• Each of the vertices lies in a different quadrant.

What are the coordinates of the vertices of your rectangle? Explain how you know the perimeter of your rectangle is

42 units

Answer:

[tex]A = (5,3)[/tex]

[tex]B = (-5,3)[/tex]

[tex]C = (-5,-8)[/tex]

[tex]D = (5,-8)[/tex]

Step-by-step explanation:

Required

[tex]Perimeter = 42[/tex]

Construct a rectangle whose perimeter is 42 units and satisfies the given conditions.

First, name the rectangle ABCD.

Such that:

[tex]A = (x_1,y_1)[/tex]

[tex]B = (x_2,y_2)[/tex]

[tex]C = (x_3,y_3)[/tex]

[tex]D = (x_4,y_4)[/tex]

For the rectangle to be either horizontal or vertical, then:

[tex]y_1 = y_2[/tex] and [tex]y_3 = y_4[/tex]

We have that:

[tex]Perimeter = 42[/tex]

Replace perimeter with its formula

[tex]2(AB + BC) = 42[/tex]

Divide both sides by 2

[tex]AB + BC = 21[/tex]

This implies that, the distance between adjacent sides (through the edges) must be equal to 21

Having said that: a set of coordinates that satisfy the given conditions are:

[tex]A = (5,3)[/tex] -- First quadrant

[tex]B = (-5,3)[/tex] -- Second quadrant

[tex]C = (-5,-8)[/tex] -- Third quadrant

[tex]D = (5,-8)[/tex] -- Fourth quadrant

The above quadrants satisfy the condition:

[tex]y_1 = y_2[/tex] and [tex]y_3 = y_4[/tex]

HOW TO KNOW THE PERIMETER IS 42

To do this, we simply calculate the distance between the edges and add them up

Distance is calculated as:

[tex]D = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2[/tex]

For AB

[tex]A = (5,3)[/tex]

[tex]B = (-5,3)[/tex]

[tex]D_1 = \sqrt{(5 - (-5))^2 + (3 - 3)^2}= \sqrt{(10)^2 + (0)^2} = \sqrt{100} = 10[/tex]

For BC

[tex]B = (-5,3)[/tex]

[tex]C = (-5,-8)[/tex]

[tex]D_2 = \sqrt{(-5 - (-5))^2 + (3 - (-8))^2}= \sqrt{(0)^2 + (11)^2} = \sqrt{121} = 11[/tex]

For CD

[tex]C = (-5,-8)[/tex]

[tex]D = (5,-8)[/tex]

[tex]D_3 = \sqrt{(-5 -5)^2 + (-8 - (-8))^2}= \sqrt{(-10)^2 + (0)^2} = \sqrt{100} = 10[/tex]

For DA

[tex]D = (5,-8)[/tex]

[tex]A = (5,3)[/tex]

[tex]D_4 = \sqrt{(5 -5)^2 + (-8 -3)^2}= \sqrt{(0)^2 + (11)^2} = \sqrt{121} = 11[/tex]

So, the perimeter is:

[tex]P = D_1 + D_2 + D_3 + D_4[/tex]

[tex]P = 10 + 11 + 10 +11[/tex]

[tex]P = 42[/tex]

See attachment for rectangle

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