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Use the moving and additivity principles about area to determine the area of the triangle in the figure in two different ways. In both cases, do not use a formula for the area of a triangle.

Use the moving and additivity principles about area to determine the area of the triangle in the figure in two different ways. In both cases, do not use a formula for the area of a triangle.

Algebra and Trigonometry (6th Edition)
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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### Exploring Area Using Moving and Additivity Principles **Objective**: Determine the area of a triangle using the moving and additivity principles, without using the traditional formula for the area of a triangle. **Instructions**: Use the concepts of moving parts and additivity to deduce the area of the given triangle. Refrain from using the familiar formula for the area of a triangle (i.e., ½ * base * height). **Diagram Description**: The diagram shows a purple shaded triangle placed on a grid. The base of the triangle spans from the third to the seventh horizontal grid line (inclusive), and the height of the triangle reaches up to the fourth vertical grid line. ### Steps for Calculating the Triangle's Area: **Method 1**: **Decomposition and Rearrangement** 1. **Decompose the Triangle**: - Observe that the triangle can be split into smaller shapes, such as right triangles or rectangles, which can be rearranged into simpler sections. - By visualising or sketching it, this approach allows us to break down the triangle into calculable parts. 2. **Rearrange the Parts**: - Move or configure the split sections to form a known shape such as a rectangle or smaller triangles. - Calculate the total combined area of these known shapes to determine the area of the original triangle. **Method 2**: **Additivity Principle** 1. **Identify a Larger Shape**: - Imagine the triangle inscribed within a parallelogram or rectangle that it completely fits within. - The imagined shape should have easily measurable areas. 2. **Subtract Excess Areas**: - Calculate the area of the larger enclosing shape. - Identify and calculate the areas of the parts of the shape that are not part of the triangle (usually other triangles or rectangles). - Subtract these excess areas from the total area of the larger shape. ### Conclusion Both methods will lead to an understanding and calculation of the triangle's area without relying on the basic area formula. This not only reinforces the understanding of geometric principles but also encourages creative problem-solving. ### Visual Representation ![Triangle on Grid](image_url) *Image depicting a purple shaded triangle inside a grid. The triangle's base is horizontal, and its height reaches the fourth vertical line.* Keep practicing these methods with various shapes to deepen your grasp of geometric concepts and their practical applications!
Transcribed Image Text:### Exploring Area Using Moving and Additivity Principles **Objective**: Determine the area of a triangle using the moving and additivity principles, without using the traditional formula for the area of a triangle. **Instructions**: Use the concepts of moving parts and additivity to deduce the area of the given triangle. Refrain from using the familiar formula for the area of a triangle (i.e., ½ * base * height). **Diagram Description**: The diagram shows a purple shaded triangle placed on a grid. The base of the triangle spans from the third to the seventh horizontal grid line (inclusive), and the height of the triangle reaches up to the fourth vertical grid line. ### Steps for Calculating the Triangle's Area: **Method 1**: **Decomposition and Rearrangement** 1. **Decompose the Triangle**: - Observe that the triangle can be split into smaller shapes, such as right triangles or rectangles, which can be rearranged into simpler sections. - By visualising or sketching it, this approach allows us to break down the triangle into calculable parts. 2. **Rearrange the Parts**: - Move or configure the split sections to form a known shape such as a rectangle or smaller triangles. - Calculate the total combined area of these known shapes to determine the area of the original triangle. **Method 2**: **Additivity Principle** 1. **Identify a Larger Shape**: - Imagine the triangle inscribed within a parallelogram or rectangle that it completely fits within. - The imagined shape should have easily measurable areas. 2. **Subtract Excess Areas**: - Calculate the area of the larger enclosing shape. - Identify and calculate the areas of the parts of the shape that are not part of the triangle (usually other triangles or rectangles). - Subtract these excess areas from the total area of the larger shape. ### Conclusion Both methods will lead to an understanding and calculation of the triangle's area without relying on the basic area formula. This not only reinforces the understanding of geometric principles but also encourages creative problem-solving. ### Visual Representation ![Triangle on Grid](image_url) *Image depicting a purple shaded triangle inside a grid. The triangle's base is horizontal, and its height reaches the fourth vertical line.* Keep practicing these methods with various shapes to deepen your grasp of geometric concepts and their practical applications!
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