Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN: 9780134463216
Author: Robert F. Blitzer
Publisher: PEARSON
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### 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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Transcribed Image Text:### 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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Step 2: Calculation of area of the figure
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Step 3: Calculation of area in second way
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Step 4: Simplification of both the expressions
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