### Problem Statement (Educational Context)2. **Objective:** - Find two expressions for the area of the following irregular figure. - Assume each of the variables \( w \), \( x \), \( y \), and \( z \) express the length of the indicated sides in units. - Show that your two expressions are equivalent to each other.**Diagram Explanation:**- The figure provided is an irregular shape resembling a sideways "E".- The lengths of the sides are labeled with variables: - Vertical length on the left side is \( z \). - The horizontal length on the top is \( y \). - There are two horizontal segments labeled \( x \), each positioned in the middle and bottom parts of the shape. - The vertical middle segment that cuts into the figure from the right is labeled \( w \).**Solution Approach:**1. **Decomposing the Figure into Rectangles:** To find the area, the irregular figure can be decomposed into simpler rectangles. - **Rectangle 1 (Top):** - Dimensions: \( y \) (horizontal) by \( x \) (vertical) - Area = \( y \times x \) - **Rectangle 2 (Middle):** - Dimensions: \( w \) (horizontal) by \( x \) (vertical) - Area = \( w \times x \) - **Rectangle 3 (Bottom):** - Dimensions: \( (y - w) \) (horizontal) by \( x \) (vertical) - Note: Since the total vertical distance from the top to the bottom is \( z \) and is composed of \( x + x \) (height of both rectangles 1 and 2), we get: \( z = x + x \). Therefore, height of rectangle 3 would be \( z - 2x \). - Assuming \( y = z - x \) (Considering the missing vertical segment) - Area = \( (y - w) \times x\)2. **Total Area:** Combine all rectangles' areas to determine the total area of the irregular figure: - Total Area = \( (y \times x) + (w \times x) + ((y - w) \times x) \) 3. **Simplify the Expression:** Simplified expression: \[

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2. Find two expressions for the area of the following irregular figure. Assume each of the variables w, x, y, and z express the length of the indicated sides in units. Show that your two expressions are equivalent to each other. y X [국 W Z X

2. Find two expressions for the area of the following irregular figure. Assume each of the variables w, x, y, and z express the length of the indicated sides in units. Show that your two expressions are equivalent to each other. y X [국 W Z X

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Question

### Problem Statement (Educational Context)

2. **Objective:**
- Find two expressions for the area of the following irregular figure.
- Assume each of the variables \( w \), \( x \), \( y \), and \( z \) express the length of the indicated sides in units.
- Show that your two expressions are equivalent to each other.

**Diagram Explanation:**
- The figure provided is an irregular shape resembling a sideways "E".
- The lengths of the sides are labeled with variables:
- Vertical length on the left side is \( z \).
- The horizontal length on the top is \( y \).
- There are two horizontal segments labeled \( x \), each positioned in the middle and bottom parts of the shape.
- The vertical middle segment that cuts into the figure from the right is labeled \( w \).

**Solution Approach:**

1. **Decomposing the Figure into Rectangles:**

To find the area, the irregular figure can be decomposed into simpler rectangles.

- **Rectangle 1 (Top):**
- Dimensions: \( y \) (horizontal) by \( x \) (vertical)
- Area = \( y \times x \)

- **Rectangle 2 (Middle):**
- Dimensions: \( w \) (horizontal) by \( x \) (vertical)
- Area = \( w \times x \)

- **Rectangle 3 (Bottom):**
- Dimensions: \( (y - w) \) (horizontal) by \( x \) (vertical)
- Note: Since the total vertical distance from the top to the bottom is \( z \) and is composed of \( x + x \) (height of both rectangles 1 and 2), we get: \( z = x + x \). Therefore, height of rectangle 3 would be \( z - 2x \).
- Assuming \( y = z - x \) (Considering the missing vertical segment)
- Area = \( (y - w) \times x\)

2. **Total Area:**

Combine all rectangles' areas to determine the total area of the irregular figure:

- Total Area = \( (y \times x) + (w \times x) + ((y - w) \times x) \)

3. **Simplify the Expression:**

Simplified expression:
\[

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