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:

1. ms. dolphin left her home, with coordinates (5, 2), and drove to the library at coordinates (5, -3). then drove to the acme with coordinates (-5, -3). if each unit represents 1 mile, what was the total distance (in miles) that ms. dolphin drove?

Explanation:

Step1: Find the fourth vertex

In a rectangle, opposite sides are equal and parallel. Given three vertices \((-4,3)\), \((1,3)\), and \((-4,-2)\).
For the \(x -\)coordinate: Since one side has \(x\) - values \(-4\) and \(1\), and another side has \(x=-4\) for two points. The fourth \(x\) - coordinate should be \(1\).
For the \(y -\)coordinate: Since one side has \(y = 3\) and \(y=-2\), the fourth \(y\) - coordinate should be \(-2\). So the fourth vertex is \((1,-2)\).

Step2: Find the length of each side

For side 1 (horizontal side with \(y = 3\)): Using the distance formula \(d=\vert x_2 - x_1\vert\). Let \(x_1=-4\), \(x_2 = 1\), then \(d=\vert1-(-4)\vert=5\).
For side 2 (vertical side with \(x=-4\)): Using the distance formula \(d=\vert y_2 - y_1\vert\). Let \(y_1 = 3\), \(y_2=-2\), then \(d=\vert-2 - 3\vert=5\).
For side 3 (horizontal side with \(y=-2\)): Using the distance formula \(d=\vert x_2 - x_1\vert\). Let \(x_1=-4\), \(x_2 = 1\), then \(d=\vert1-(-4)\vert=5\).
For side 4 (vertical side with \(x = 1\)): Using the distance formula \(d=\vert y_2 - y_1\vert\). Let \(y_1 = 3\), \(y_2=-2\), then \(d=\vert-2 - 3\vert=5\).

Step3: Add the length of all 4 sides

Perimeter \(P=5 + 5+5 + 5=20\)

Answer:

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