Explain how to use a coordinate plane to find the area of a rectangle with vertices $(-4,9),(-4,-3),(-1,-3)$, and $(-1,9)$. What is the area?
Answer (1)
Plot the vertices ( − 4 , 9 ) , ( − 4 , − 3 ) , ( − 1 , − 3 ) , and ( − 1 , 9 ) on the coordinate plane.
Calculate the length of the rectangle: l e n g t h = ∣ − 1 − ( − 4 ) ∣ = 3 .
Calculate the width of the rectangle: w i d t h = ∣9 − ( − 3 ) ∣ = 12 .
Determine the area of the rectangle: A re a = l e n g t h × w i d t h = 3 × 12 = 36 . The area is 36 .
Explanation
Plot the vertices First, let's plot the given vertices of the rectangle on the coordinate plane: ( − 4 , 9 ) , ( − 4 , − 3 ) , ( − 1 , − 3 ) , and ( − 1 , 9 ) . Visualizing these points helps us understand the rectangle's dimensions.
Find the length Next, we need to find the length and width of the rectangle. We can find the length by calculating the distance between the points ( − 4 , 9 ) and ( − 1 , 9 ) . Since these points have the same y -coordinate, the length is simply the absolute difference of their x -coordinates:
Calculate the length l e n g t h = ∣ − 1 − ( − 4 ) ∣ = ∣ − 1 + 4∣ = ∣3∣ = 3
Find the width Similarly, we can find the width by calculating the distance between the points ( − 4 , 9 ) and ( − 4 , − 3 ) . Since these points have the same x -coordinate, the width is the absolute difference of their y -coordinates:
Calculate the width w i d t h = ∣9 − ( − 3 ) ∣ = ∣9 + 3∣ = ∣12∣ = 12
Calculate the area Now that we have the length and width, we can calculate the area of the rectangle using the formula:
Area calculation A re a = l e n g t h × w i d t h = 3 × 12 = 36
State the final answer Therefore, the area of the rectangle is 36 square units.
Examples
Understanding area calculations is crucial in many real-world scenarios. For instance, when designing a room layout, you need to calculate the area of the room to determine how much flooring to purchase. Similarly, architects use area calculations to plan buildings, and farmers use them to determine the amount of land needed for planting crops. Knowing how to find the area of basic shapes like rectangles is a fundamental skill with broad applications.
