A Rectangle Has Vertices At \[$(-1,6), (-1,-2), (3,6)\$\], And \[$(3,-2)\$\]. Sara Says The Area Of The Rectangle Is 16 Square Units And Her Work Is Shown Below.$\[ \begin{tabular}{|c|c|} \hline Steps & Sara's Work \\ \hline Step 1

Mar 1, 2025 by ADMIN 232 views

$A rectangle has vertices at ${$(-1,6), (-1,-2), (3,6)\$}$, and ${$(3,-2)\$}$. Sara says the area of the rectangle is 16 square units and her work is shown below.$

Introduction

Understanding the Problem In mathematics, a rectangle is a type of quadrilateral with four right angles and opposite sides of equal length. Given the vertices of a rectangle, we can calculate its area using various methods. In this article, we will examine Sara's work and determine if her calculation of the area is correct.

Sara's Work

Step 1: Find the Length of the Base

Sara starts by finding the length of the base of the rectangle, which is the horizontal distance between the points (-1,6) and (3,6). She calculates this distance as follows:

base_length = abs(3 - (-1))
print(base_length)

This code calculates the absolute value of the difference between the x-coordinates of the two points, giving a base length of 4 units.

Step 2: Find the Height of the Rectangle

Next, Sara finds the height of the rectangle, which is the vertical distance between the points (-1,6) and (-1,-2). She calculates this distance as follows:

height = abs(6 - (-2))
print(height)

This code calculates the absolute value of the difference between the y-coordinates of the two points, giving a height of 8 units.

Step 3: Calculate the Area of the Rectangle

Finally, Sara calculates the area of the rectangle by multiplying the base length and the height:

area = base_length * height
print(area)

This code multiplies the base length and the height, giving an area of 32 square units.

Analysis of Sara's Work

Sara's work appears to be correct, but her final answer is different from the one she initially stated. Let's re-examine her calculations to see if there are any errors.

Re-examining the Base Length

Sara's calculation of the base length is correct, but let's re-examine it to make sure. The base length is the horizontal distance between the points (-1,6) and (3,6). We can calculate this distance using the distance formula:

import math
base_length = math.sqrt((3 - (-1))**2 + (6 - 6)**2)
print(base_length)

This code calculates the distance between the two points using the distance formula, giving a base length of 4 units.

Re-examining the Height

Sara's calculation of the height is also correct, but let's re-examine it to make sure. The height is the vertical distance between the points (-1,6) and (-1,-2). We can calculate this distance using the distance formula:

import math
height = math.sqrt((-1 - (-1))**2 + (6 - (-2))**2)
print(height)

This code calculates the distance between the two points using the distance formula, giving a height of 8 units.

Re-examining the Area

Finally, let's re-examine Sara's calculation of the area. We can calculate the area by multiplying the base length and the height:

area = base_length * height
print(area)

This code multiplies the base length and the height, giving an area of 32 square units.

Conclusion

In conclusion, Sara's work appears to be correct, but her final answer is different from the one she initially stated. After re-examining her calculations, we found that her base length and height are correct, but her area is incorrect. The correct area of the rectangle is 32 square units, not 16 square units.

Frequently Asked Questions

Q: What is the formula for calculating the area of a rectangle?

A: The formula for calculating the area of a rectangle is length × width.

Q: How do I calculate the length and width of a rectangle?

A: To calculate the length and width of a rectangle, you can use the distance formula: √((x2 - x1)^2 + (y2 - y1)^2).

Q: What is the difference between the area and the perimeter of a rectangle?

A: The area of a rectangle is the product of its length and width, while the perimeter is the sum of all its sides.

References

[1] "Rectangle." Wikipedia, Wikimedia Foundation, 2023, en.wikipedia.org/wiki/Rectangle.
[2] "Distance Formula." Math Open Reference, mathopenref.com/distance.html.

Additional Resources

[1] "Geometry." Khan Academy, khanacademy.org/math/geometry.
[2] "Mathematics." Wolfram Alpha, www.wolframalpha.com.

Final Answer

The final answer is: $\boxed{32}$

Q&A: Understanding the Problem

Q: What is a rectangle?

A: A rectangle is a type of quadrilateral with four right angles and opposite sides of equal length.

Q: How do I find the area of a rectangle?

A: To find the area of a rectangle, you need to multiply the length and width of the rectangle.

Q: What is the formula for calculating the area of a rectangle?

A: The formula for calculating the area of a rectangle is length × width.

Q: How do I calculate the length and width of a rectangle?

A: To calculate the length and width of a rectangle, you can use the distance formula: √((x2 - x1)^2 + (y2 - y1)^2).

Q: What is the difference between the area and the perimeter of a rectangle?

A: The area of a rectangle is the product of its length and width, while the perimeter is the sum of all its sides.

Q: How do I find the perimeter of a rectangle?

A: To find the perimeter of a rectangle, you need to add up the lengths of all its sides.

Q: What is the formula for calculating the perimeter of a rectangle?

A: The formula for calculating the perimeter of a rectangle is 2 × (length + width).

Q: Can I use the distance formula to find the length and width of a rectangle?

A: Yes, you can use the distance formula to find the length and width of a rectangle.

Q: What is the difference between the length and width of a rectangle?

A: The length and width of a rectangle are the two dimensions that make up the rectangle.

Q: How do I find the length and width of a rectangle if I know the coordinates of its vertices?

A: To find the length and width of a rectangle if you know the coordinates of its vertices, you can use the distance formula.

Q&A: Calculating the Area of a Rectangle

Q: How do I calculate the area of a rectangle if I know the length and width?

A: To calculate the area of a rectangle if you know the length and width, you can multiply the length and width together.

Q: What is the formula for calculating the area of a rectangle if I know the length and width?

A: The formula for calculating the area of a rectangle if you know the length and width is length × width.

Q: Can I use the distance formula to calculate the area of a rectangle?

A: No, you cannot use the distance formula to calculate the area of a rectangle.

Q: What is the difference between the area and the perimeter of a rectangle?

A: The area of a rectangle is the product of its length and width, while the perimeter is the sum of all its sides.

Q&A: Real-World Applications of Rectangles

Q: Where do rectangles appear in real life?

A: Rectangles appear in many real-life situations, such as in the design of buildings, bridges, and other structures.

Q: How do rectangles relate to other geometric shapes?

A: Rectangles are a type of quadrilateral, and they can be combined with other geometric shapes to form more complex shapes.

Q: Can rectangles be used to solve real-world problems?

A: Yes, rectangles can be used to solve real-world problems, such as designing a room or a building.

Q&A: Common Mistakes When Calculating the Area of a Rectangle

Q: What are some common mistakes people make when calculating the area of a rectangle?

A: Some common mistakes people make when calculating the area of a rectangle include:

Forgetting to multiply the length and width
Using the wrong formula for calculating the area
Not using the correct units for the length and width

Q: How can I avoid making these mistakes?

A: To avoid making these mistakes, you can:

Double-check your calculations
Use the correct formula for calculating the area
Make sure to use the correct units for the length and width

Q&A: Conclusion

Q: What is the final answer to the problem?

A: The final answer to the problem is 32 square units.

Q: What is the main takeaway from this article?

A: The main takeaway from this article is that rectangles are a type of quadrilateral with four right angles and opposite sides of equal length, and that the area of a rectangle can be calculated by multiplying the length and width.

Q: What are some common applications of rectangles in real life?

A: Some common applications of rectangles in real life include designing buildings, bridges, and other structures.

Q: How can I use rectangles to solve real-world problems?

A: You can use rectangles to solve real-world problems by designing a room or a building, or by calculating the area of a rectangle to determine the amount of materials needed for a project.