Rectangle Area: Correcting Pablo's Calculation

Let's dive into the world of rectangles and areas, and help our friend Pablo correct his calculation! We have a rectangle with vertices at $(-2,11)$, $(-2,4)$, $(6,11)$, and $(6,4)$. Pablo believes the area of this rectangle is 49 square units, and his base calculation is $|-2| + |6| = 8$. Let's see where he went wrong and find the correct area.

Understanding the Rectangle

Before we analyze Pablo's work, let's visualize and understand the rectangle we're dealing with. The vertices provided define the corners of the rectangle in the Cartesian plane. We have:

• Vertex 1: $(-2, 11)$
• Vertex 2: $(-2, 4)$
• Vertex 3: $(6, 11)$
• Vertex 4: $(6, 4)$

By plotting these points, or even just visualizing them, we can determine the lengths of the sides of the rectangle. These lengths are essential for calculating the area.

Calculating the Base and Height

To find the area of a rectangle, we need its base and height. The base is the length of the horizontal side, and the height is the length of the vertical side.

Finding the Base

The base is the distance between the points $(-2, 11)$ and $(6, 11)$ (or between $(-2, 4)$ and $(6, 4)$). Since the y-coordinates are the same, we can find the length by taking the absolute difference of the x-coordinates:

Base = $|6 - (-2)| = |6 + 2| = |8| = 8$

So, the base of the rectangle is 8 units. Pablo correctly calculated this.

Finding the Height

The height is the distance between the points $(-2, 11)$ and $(-2, 4)$ (or between $(6, 11)$ and $(6, 4)$). Since the x-coordinates are the same, we can find the length by taking the absolute difference of the y-coordinates:

Height = $|11 - 4| = |7| = 7$

So, the height of the rectangle is 7 units.

Calculating the Area

Now that we have the base and height, we can calculate the area of the rectangle using the formula:

Area = Base × Height

Area = $8 \;×\; 7 = 56$

Therefore, the area of the rectangle is 56 square units.

Analyzing Pablo's Error

Pablo claims the area is 49 square units and his base calculation is $|-2| + |6| = 8$, which is coincidentally correct for the base. However, we need to understand where the error in his overall calculation lies. The base calculation itself isn't the problem, since he correctly found the length of the base to be 8. The issue isn't explicitly stated, but we can infer that Pablo likely made a mistake in determining the height or in the final area calculation itself. He might have incorrectly calculated the height, or perhaps he used the wrong formula or made an arithmetic error when multiplying. The correct height is 7, and the correct area is 56.

Correcting Pablo's Work

Let's break down the correct steps to find the area:

1. Identify the coordinates: $(-2,11), (-2,4), (6,11)$, and $(6,4)$.
2. Calculate the base: The base is the difference in x-coordinates: $|6 - (-2)| = 8$.
3. Calculate the height: The height is the difference in y-coordinates: $|11 - 4| = 7$.
4. Calculate the area: Area = Base × Height = $8 \;×\; 7 = 56$.

Therefore, the area of the rectangle is 56 square units, not 49 square units.

Key Takeaways

• Understanding Coordinates: The coordinates of the vertices are essential for determining the dimensions of the rectangle.
• Base and Height: Correctly identifying and calculating the base and height is crucial for finding the area.
• Area Formula: The area of a rectangle is found by multiplying its base and height.
• Attention to Detail: Accuracy in calculations is vital to avoid errors.

In conclusion, the correct area of the rectangle is 56 square units. Pablo made an error in his overall calculation, likely in determining the height or in the final area calculation itself. By carefully calculating the base and height and applying the area formula, we can arrive at the correct answer. Always double-check your work to ensure accuracy in mathematical problems!

For more information on calculating the area of rectangles and other geometric shapes, you can visit Khan Academy's geometry section.