Question

1. three coordinates of a rectangle are (-4, 3), (1, 3), and (-4, -2).

a. what are the coordinates of the fourth (corner) vertex?
b. find the length of each side. side 1:__, side 2:, side 3:, side 4:__
c. add the length of all 4 sides __+++__=
d. the perimeter of the rectangle is:____

Explanation:

Step1: Find the fourth vertex

In a rectangle, opposite sides are equal and parallel.
For the \(x -\)coordinate:
We have two points with \(x=-4\) and one with \(x = 1\). The fourth vertex's \(x -\)coordinate should be \(1\).
For the \(y -\)coordinate:
We have two points with \(y = 3\) and one with \(y=-2\). The fourth vertex's \(y -\)coordinate should be \(-2\).
So the fourth vertex is \((1,-2)\)

Step2: Find the length of each side

Use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)
Side 1: Between \((-4,3)\) and \((1,3)\)
\(d_1=\sqrt{(1-(-4))^2+(3 - 3)^2}=\sqrt{(5)^2+0}=5\)
Side 2: Between \((-4,3)\) and \((-4,-2)\)
\(d_2=\sqrt{(-4-(-4))^2+(-2 - 3)^2}=\sqrt{0+(-5)^2}=5\)
Side 3: Between \((1,3)\) and \((1,-2)\)
\(d_3=\sqrt{(1 - 1)^2+(-2 - 3)^2}=\sqrt{0+(-5)^2}=5\)
Side 4: Between \((1,-2)\) and \((-4,-2)\)
\(d_4=\sqrt{(-4 - 1)^2+(-2+2)^2}=\sqrt{(-5)^2+0}=5\)

Step3: Add the length of all 4 sides

Perimeter \(P=d_1 + d_2+d_3 + d_4=5 + 5+5 + 5=20\)

Answer:

a. \((1,-2)\)
b. Side 1: \(5\), Side 2: \(5\), Side 3: \(5\), Side 4: \(5\)
c. \(5+5+5 + 5\)
d. \(20\)