Question

1. (inv. 3) the coordinates of three vertices of a rectangle are (3, 2), (3, -3), and (-1, -3).

a. what are the coordinates of the fourth vertex?
b. what is the area of the rectangle?

Explanation:

Part a

Step1: Analyze rectangle properties

In a rectangle, opposite sides are equal and parallel (so same \(x\)- or \(y\)-coordinates for vertical/horizontal sides). Let's label the points: \(A(3, 2)\), \(B(3, -3)\), \(C(-1, -3)\).

Step2: Find vertical/horizontal pairs

• \(A\) and \(B\) have the same \(x\)-coordinate (\(x = 3\)), so this is a vertical side. Length: \(|2 - (-3)| = 5\).
• \(B\) and \(C\) have the same \(y\)-coordinate (\(y = -3\)), so this is a horizontal side. Length: \(|3 - (-1)| = 4\).

Step3: Determine the fourth vertex

The fourth vertex \(D\) should have the same \(x\)-coordinate as \(C\) (\(x = -1\)) and the same \(y\)-coordinate as \(A\) (\(y = 2\)) (to complete the rectangle: opposite sides parallel). So \(D(-1, 2)\).

Part b

Step1: Recall rectangle area formula

Area of a rectangle is \( \text{length} \times \text{width} \).

Step2: Identify length and width

From part a, length (vertical side) is \(5\) (distance between \(A\) and \(B\): \(|2 - (-3)| = 5\)), width (horizontal side) is \(4\) (distance between \(B\) and \(C\): \(|3 - (-1)| = 4\)).

Step3: Calculate area

Multiply length and width: \( 5 \times 4 = 20 \).

Part a Answer: \((-1, 2)\)

Part b Answer: \(20\)

Answer:

Step1: Recall rectangle area formula

Area of a rectangle is \( \text{length} \times \text{width} \).

Step2: Identify length and width

From part a, length (vertical side) is \(5\) (distance between \(A\) and \(B\): \(|2 - (-3)| = 5\)), width (horizontal side) is \(4\) (distance between \(B\) and \(C\): \(|3 - (-1)| = 4\)).

Step3: Calculate area

Multiply length and width: \( 5 \times 4 = 20 \).

Part a Answer: \((-1, 2)\)

Part b Answer: \(20\)