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
Related questions
Question
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**Finding the Centroid of a Composite Shape**
**Objective:** Determine the centroid (\(\overline{x}\) and \(\overline{y}\)) of the given composite shape with respect to the shown axes.
**Description of the Shape:**
The diagram represents a composite shape consisting of a rectangle and a right triangle, as well as an additional rectangle extending from the bottom left corner.
1. **Left Rectangle**:
- Width: 2 inches
- Height: 2 inches
- Positioned at the bottom left corner adjacent to the axis
2. **Middle Rectangle**:
- Width: 4 inches
- Height: 4 inches
- Stacked on top of the left rectangle, sharing its left side, extending upwards from 2 inches to 6 inches
3. **Right Triangle**:
- Base: 6 inches
- Height: 4 inches
- Positioned to the right of the middle rectangle, with the base along the x-axis and merging with the top side of the middle rectangle
**Approach for Finding the Centroid:**
To find the centroid of a composite shape:
1. Divide the shape into basic geometric shapes (rectangles and triangles).
2. Calculate the area of each shape.
3. Determine the centroid of each separate shape in terms of x and y coordinates, relative to the given axes.
4. Use the formula for composite centroids:
\[
\overline{x} = \frac{\sum (x_i \cdot A_i)}{\sum A_i}, \quad \overline{y} = \frac{\sum (y_i \cdot A_i)}{\sum A_i}
\]
Where \(x_i\) and \(y_i\) are the centroids of the individual shapes, and \(A_i\) are their respective areas.
By calculating and summing the moments of the areas about the axes, the coordinates of the centroid of the entire shape can be determined.
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Transcribed Image Text:---
**Finding the Centroid of a Composite Shape**
**Objective:** Determine the centroid (\(\overline{x}\) and \(\overline{y}\)) of the given composite shape with respect to the shown axes.
**Description of the Shape:**
The diagram represents a composite shape consisting of a rectangle and a right triangle, as well as an additional rectangle extending from the bottom left corner.
1. **Left Rectangle**:
- Width: 2 inches
- Height: 2 inches
- Positioned at the bottom left corner adjacent to the axis
2. **Middle Rectangle**:
- Width: 4 inches
- Height: 4 inches
- Stacked on top of the left rectangle, sharing its left side, extending upwards from 2 inches to 6 inches
3. **Right Triangle**:
- Base: 6 inches
- Height: 4 inches
- Positioned to the right of the middle rectangle, with the base along the x-axis and merging with the top side of the middle rectangle
**Approach for Finding the Centroid:**
To find the centroid of a composite shape:
1. Divide the shape into basic geometric shapes (rectangles and triangles).
2. Calculate the area of each shape.
3. Determine the centroid of each separate shape in terms of x and y coordinates, relative to the given axes.
4. Use the formula for composite centroids:
\[
\overline{x} = \frac{\sum (x_i \cdot A_i)}{\sum A_i}, \quad \overline{y} = \frac{\sum (y_i \cdot A_i)}{\sum A_i}
\]
Where \(x_i\) and \(y_i\) are the centroids of the individual shapes, and \(A_i\) are their respective areas.
By calculating and summing the moments of the areas about the axes, the coordinates of the centroid of the entire shape can be determined.
---
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