To Find The Area Of Parallelogram RSTU, Juan Starts By Drawing A Rectangle Around It. Each Vertex Of Parallelogram RSTU Is On A Side Of The Rectangle He Draws.Which Expression Can Be Subtracted From The Area Of The Rectangle To Find The Area Of

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Introduction

In geometry, finding the area of a parallelogram can be a challenging task, especially when dealing with complex shapes. However, with the help of a rectangle, we can simplify the process and find the area of the parallelogram. In this article, we will explore how to find the area of a parallelogram by drawing a rectangle around it and subtracting a specific expression from the area of the rectangle.

Understanding the Problem

Let's consider the problem presented by Juan, where he draws a rectangle around the parallelogram RSTU. Each vertex of the parallelogram is on a side of the rectangle he draws. The goal is to find the area of the parallelogram by subtracting a specific expression from the area of the rectangle.

Drawing a Rectangle Around the Parallelogram

To find the area of the parallelogram, Juan draws a rectangle around it. The rectangle has the same base and height as the parallelogram. Let's denote the base of the parallelogram as b and the height as h. The area of the rectangle is given by the formula:

Area of rectangle = base × height

Area of rectangle = b × h

Finding the Area of the Parallelogram

The area of the parallelogram can be found by subtracting a specific expression from the area of the rectangle. To do this, we need to understand the relationship between the area of the rectangle and the area of the parallelogram.

The Relationship Between the Area of the Rectangle and the Parallelogram

The area of the parallelogram is equal to the area of the rectangle minus the areas of the two triangles formed by the rectangle and the parallelogram. Let's denote the area of each triangle as Δ. The area of the parallelogram can be expressed as:

Area of parallelogram = Area of rectangle - 2 × Δ

Finding the Expression to Subtract

To find the expression to subtract from the area of the rectangle, we need to find the area of each triangle Δ. The area of each triangle is given by the formula:

Δ = (base × height) / 2

Δ = (b × h) / 2

Subtracting the Expression from the Area of the Rectangle

Now that we have found the area of each triangle Δ, we can subtract it from the area of the rectangle to find the area of the parallelogram. The expression to subtract is:

2 × Δ = 2 × (b × h) / 2

2 × Δ = b × h

Conclusion

In conclusion, to find the area of a parallelogram using a rectangle, we need to draw a rectangle around the parallelogram and subtract a specific expression from the area of the rectangle. The expression to subtract is the area of the two triangles formed by the rectangle and the parallelogram. By following these steps, we can find the area of the parallelogram with ease.

Example

Let's consider an example to illustrate the concept. Suppose we have a parallelogram with a base of 6 units and a height of 4 units. We draw a rectangle around the parallelogram with the same base and height. The area of the rectangle is:

Area of rectangle = b × h

Area of rectangle = 6 × 4

Area of rectangle = 24

The area of each triangle Δ is:

Δ = (base × height) / 2

Δ = (6 × 4) / 2

Δ = 12

The expression to subtract from the area of the rectangle is:

2 × Δ = 2 × 12

2 × Δ = 24

The area of the parallelogram is:

Area of parallelogram = Area of rectangle - 2 × Δ

Area of parallelogram = 24 - 24

Area of parallelogram = 0

Conclusion

In this article, we have explored how to find the area of a parallelogram using a rectangle. We have seen that the area of the parallelogram can be found by subtracting a specific expression from the area of the rectangle. The expression to subtract is the area of the two triangles formed by the rectangle and the parallelogram. By following these steps, we can find the area of the parallelogram with ease.

Key Takeaways

  • To find the area of a parallelogram using a rectangle, we need to draw a rectangle around the parallelogram and subtract a specific expression from the area of the rectangle.
  • The expression to subtract is the area of the two triangles formed by the rectangle and the parallelogram.
  • The area of each triangle is given by the formula: Δ = (base × height) / 2.
  • The area of the parallelogram is equal to the area of the rectangle minus the areas of the two triangles formed by the rectangle and the parallelogram.

Final Thoughts

Q: What is the relationship between the area of a rectangle and the area of a parallelogram?

A: The area of a parallelogram is equal to the area of a rectangle minus the areas of the two triangles formed by the rectangle and the parallelogram.

Q: How do I find the area of each triangle formed by the rectangle and the parallelogram?

A: To find the area of each triangle, you need to use the formula: Δ = (base × height) / 2. This formula calculates the area of each triangle by multiplying the base and height of the triangle and then dividing the result by 2.

Q: What is the expression to subtract from the area of the rectangle to find the area of the parallelogram?

A: The expression to subtract from the area of the rectangle is the area of the two triangles formed by the rectangle and the parallelogram. This expression is equal to 2 × Δ, where Δ is the area of each triangle.

Q: How do I find the area of the parallelogram using the expression to subtract?

A: To find the area of the parallelogram, you need to subtract the expression 2 × Δ from the area of the rectangle. This can be expressed as: Area of parallelogram = Area of rectangle - 2 × Δ.

Q: What are the key steps to find the area of a parallelogram using a rectangle?

A: The key steps to find the area of a parallelogram using a rectangle are:

  1. Draw a rectangle around the parallelogram.
  2. Find the area of the rectangle by multiplying the base and height of the rectangle.
  3. Find the area of each triangle formed by the rectangle and the parallelogram using the formula Δ = (base × height) / 2.
  4. Find the expression to subtract from the area of the rectangle by multiplying the area of each triangle by 2.
  5. Subtract the expression from the area of the rectangle to find the area of the parallelogram.

Q: What are the benefits of finding the area of a parallelogram using a rectangle?

A: The benefits of finding the area of a parallelogram using a rectangle include:

  • Simplifying the process of finding the area of a parallelogram.
  • Reducing the complexity of the problem.
  • Making it easier to visualize the problem and understand the relationship between the area of the rectangle and the area of the parallelogram.

Q: Can I use this method to find the area of any parallelogram?

A: Yes, you can use this method to find the area of any parallelogram. However, you need to make sure that the rectangle you draw around the parallelogram has the same base and height as the parallelogram.

Q: What are some common mistakes to avoid when finding the area of a parallelogram using a rectangle?

A: Some common mistakes to avoid when finding the area of a parallelogram using a rectangle include:

  • Failing to draw a rectangle around the parallelogram with the same base and height.
  • Failing to find the area of each triangle formed by the rectangle and the parallelogram.
  • Failing to find the expression to subtract from the area of the rectangle.
  • Failing to subtract the expression from the area of the rectangle to find the area of the parallelogram.

Q: How can I practice finding the area of a parallelogram using a rectangle?

A: You can practice finding the area of a parallelogram using a rectangle by:

  • Drawing rectangles around different parallelograms and finding their areas.
  • Using different values for the base and height of the rectangle and parallelogram.
  • Using different shapes and sizes of parallelograms.
  • Working with different types of problems, such as finding the area of a parallelogram with a given base and height.

Q: What are some real-world applications of finding the area of a parallelogram using a rectangle?

A: Some real-world applications of finding the area of a parallelogram using a rectangle include:

  • Architecture: Finding the area of a parallelogram can be useful in architecture when designing buildings or structures.
  • Engineering: Finding the area of a parallelogram can be useful in engineering when designing bridges or other structures.
  • Geography: Finding the area of a parallelogram can be useful in geography when studying the shape and size of different regions or countries.
  • Science: Finding the area of a parallelogram can be useful in science when studying the shape and size of different objects or structures.