Question

1. three coordinates of a rectangle are (-4,3), (7,3), 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.
If we have three vertices \((x_1,y_1)=(-4,3)\), \((x_2,y_2)=(1,3)\), \((x_3,y_3)=(-4,-2)\)
For a rectangle, if two points have the same \(y -\) coordinate (\((-4,3)\) and \((1,3)\)), and two points have the same \(x -\) coordinate (\((-4,3)\) and \((-4,-2)\))
The fourth vertex \((x,y)\) should have \(x = 1\) (same \(x -\) value as \((1,3)\)) and \(y=-2\) (same \(y -\) value as \((-4,-2)\))

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 \((1,3)\)
\(d_1=\sqrt{(1-(-4))^2+(3 - 3)^2}=\sqrt{(1 + 4)^2}=\sqrt{25}=5\)
Side 2: Between \((-4,3)\) and \((-4,-2)\)
\(d_2=\sqrt{(-4-(-4))^2+(-2 - 3)^2}=\sqrt{(0)^2+(-5)^2}=5\)
Side 3: Between \((1,3)\) and \((1,-2)\) (after finding the fourth vertex \((1,-2)\))
\(d_3=\sqrt{(1 - 1)^2+(-2 - 3)^2}=5\)
Side 4: Between \((-4,-2)\) and \((1,-2)\)
\(d_4=\sqrt{(1-(-4))^2+(-2+2)^2}=5\)

Step3: Add the length of all 4 sides

Perimeter of a rectangle \(P=2(l + w)\), but also \(P=d_1 + d_2+d_3 + d_4\)
\(P=5+5 + 5+5=20\)

Answer:

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