Plot the vertices (-2,-8), (-2, 2), and (8, 2).
What is the last vertex of the rectangle?
What is the area of the rectangle?
square units
Answer (1)
Plot the given vertices (-2, -8), (-2, 2), and (8, 2) on the coordinate plane.
Determine the coordinates of the fourth vertex by recognizing that it must align horizontally with (8, 2) and vertically with (-2, -8), giving the point (8, -8).
Calculate the length and width of the rectangle using the distance between the points: l e n g t h = ∣8 − ( − 2 ) ∣ = 10 and w i d t h = ∣2 − ( − 8 ) ∣ = 10 .
Compute the area of the rectangle by multiplying the length and width: A re a = 10 × 10 = 100 .
Explanation
Problem Analysis We are given three vertices of a rectangle: (-2, -8), (-2, 2), and (8, 2). Our goal is to find the coordinates of the fourth vertex and then calculate the area of the rectangle.
Plotting the Vertices Let's plot the given vertices on the coordinate plane. We can observe that the points (-2, 2) and (8, 2) share the same y-coordinate, which means they lie on the same horizontal line. Similarly, the points (-2, -8) and (-2, 2) share the same x-coordinate, indicating they lie on the same vertical line. This tells us that the given points form a right angle at the vertex (-2, 2).
Finding the Fourth Vertex To form a rectangle, the fourth vertex must share the x-coordinate with (8, 2) and the y-coordinate with (-2, -8). Therefore, the fourth vertex is (8, -8).
Calculating the Length Now, let's calculate the length of the horizontal side. This is the distance between the points (-2, 2) and (8, 2). We use the distance formula (or simply subtract the x-coordinates since the y-coordinates are the same): l e n g t h = ∣8 − ( − 2 ) ∣ = ∣8 + 2∣ = 10
Calculating the Width Next, we calculate the length of the vertical side. This is the distance between the points (-2, -8) and (-2, 2). We use the distance formula (or simply subtract the y-coordinates since the x-coordinates are the same): w i d t h = ∣2 − ( − 8 ) ∣ = ∣2 + 8∣ = 10
Calculating the Area Now that we have the length and width, we can calculate the area of the rectangle using the formula: A re a = l e n g t h × w i d t h = 10 × 10 = 100 So, the area of the rectangle is 100 square units.
Final Answer The fourth vertex of the rectangle is (8, -8), and the area of the rectangle is 100 square units.
Examples
Understanding area calculations is crucial in many real-world scenarios. For example, when designing a room, you need to calculate the area to determine how much flooring to buy. Similarly, architects use area calculations to plan buildings, and farmers use them to determine how much land is needed for planting crops. Knowing how to calculate the area of a rectangle helps in efficient space planning and resource management.
