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2) Figure 12.47 shows a shaded parallelogram in- side a rectangle. Use the moving and additivity principles to find an expression for the area of the shaded parallelogram in terms of all of the lengths x, y, and z. Explain your reasoning. Then simplify the expression you found. (lub ow) T ∙y 200269 Nov N

2) Figure 12.47 shows a shaded parallelogram in- side a rectangle. Use the moving and additivity principles to find an expression for the area of the shaded parallelogram in terms of all of the lengths x, y, and z. Explain your reasoning. Then simplify the expression you found. (lub ow) T ∙y 200269 Nov N

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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### Problem Statement **Figure 12.47** shows a shaded parallelogram inside a rectangle. Use the moving and additivity principles to find an expression for the area of the shaded parallelogram in terms of all of the lengths \( x \), \( y \), and \( z \). Explain your reasoning. Then simplify the expression you found. ### Diagram Explanation In the figure, a parallelogram is shaded within a rectangle. The rectangle has the following labeled dimensions: - \( x \) is the vertical height on the left side of the rectangle. - \( y \) is the horizontal length along the bottom part of the rectangle. - \( z \) is the horizontal length along the bottom part of the rectangle towards the right side, forming part of the base of the parallelogram. The shaded parallelogram's slant is such that its height is \( x \) (vertical distance) and its base spans the length from the left side (the starting vertical edge) to the end of \( z \) on the right. ### Solution Approach 1. **Identifying the Parallelogram's Dimensions:** - The height of the shaded parallelogram is \( x \). - The base of the shaded parallelogram is \( y + z \). 2. **Applying the Area Formula for a Parallelogram:** - The area \( A \) of a parallelogram is given by the formula: \[ A = \text{base} \times \text{height} \] - Substituting the expressions for base and height from the given diagram: \[ A = (y + z) \times x \] 3. **Simplified Expression:** - The simplified expression for the area of the shaded parallelogram is: \[ A = x(y + z) \] ### Conclusion By analyzing the dimensions provided in Figure 12.47 and applying the basic formula for the area of a parallelogram, we derived that the area of the shaded parallelogram can be expressed as \( x(y + z) \). This step-by-step explanation demonstrates the use of geometric principles to break down and solve the problem on finding the area in terms of the given lengths \( x \), \( y \), and \( z \).
Transcribed Image Text:### Problem Statement **Figure 12.47** shows a shaded parallelogram inside a rectangle. Use the moving and additivity principles to find an expression for the area of the shaded parallelogram in terms of all of the lengths \( x \), \( y \), and \( z \). Explain your reasoning. Then simplify the expression you found. ### Diagram Explanation In the figure, a parallelogram is shaded within a rectangle. The rectangle has the following labeled dimensions: - \( x \) is the vertical height on the left side of the rectangle. - \( y \) is the horizontal length along the bottom part of the rectangle. - \( z \) is the horizontal length along the bottom part of the rectangle towards the right side, forming part of the base of the parallelogram. The shaded parallelogram's slant is such that its height is \( x \) (vertical distance) and its base spans the length from the left side (the starting vertical edge) to the end of \( z \) on the right. ### Solution Approach 1. **Identifying the Parallelogram's Dimensions:** - The height of the shaded parallelogram is \( x \). - The base of the shaded parallelogram is \( y + z \). 2. **Applying the Area Formula for a Parallelogram:** - The area \( A \) of a parallelogram is given by the formula: \[ A = \text{base} \times \text{height} \] - Substituting the expressions for base and height from the given diagram: \[ A = (y + z) \times x \] 3. **Simplified Expression:** - The simplified expression for the area of the shaded parallelogram is: \[ A = x(y + z) \] ### Conclusion By analyzing the dimensions provided in Figure 12.47 and applying the basic formula for the area of a parallelogram, we derived that the area of the shaded parallelogram can be expressed as \( x(y + z) \). This step-by-step explanation demonstrates the use of geometric principles to break down and solve the problem on finding the area in terms of the given lengths \( x \), \( y \), and \( z \).
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