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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![### Understanding Composite Figures
In this section, we will learn how to calculate the area of composite figures. Composite figures are shapes that consist of two or more geometric figures.
#### Example Problem
Consider the composite figure shown below:
[Insert Image Here]
The figure is composed of a rectangle and a right triangle. The dimensions are as follows:
- The base of the triangle is 4 units.
- The height of the triangle (which is also the width of the rectangle) is 3 units.
- The length of the rectangle is 10 units.
Let’s break down the steps to calculate the total area of this composite figure.
### Step-by-Step Solution
1. **Find the Area of the Triangle:**
The formula for the area of a triangle is:
\[
\text{Area of a Triangle} = \frac{1}{2} \times \text{base} \times \text{height}
\]
Substitute the given values:
\[
\text{Area of the Triangle} = \frac{1}{2} \times 4 \times 3 = 6 \text{ square units}
\]
2. **Find the Area of the Rectangle:**
The formula for the area of a rectangle is:
\[
\text{Area of a Rectangle} = \text{length} \times \text{width}
\]
Substitute the given values:
\[
\text{Area of the Rectangle} = 10 \times 3 = 30 \text{ square units}
\]
3. **Calculate the Total Area of the Composite Figure:**
Add the areas of the rectangle and the triangle together:
\[
\text{Total Area} = \text{Area of the Triangle} + \text{Area of the Rectangle} = 6 + 30 = 36 \text{ square units}
\]
#### Conclusion
The total area of the composite figure is 36 square units. By breaking down the composite figure into simpler shapes, we can easily calculate the total area. This method can be applied to a variety of composite figures composed of basic geometric shapes.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F046d8025-90a9-4a04-83c7-1211a0a4be33%2F40d02eee-d41f-44b9-b866-ace0b2cf89bb%2Fjbfqfqk_processed.png&w=3840&q=75)
Transcribed Image Text:### Understanding Composite Figures
In this section, we will learn how to calculate the area of composite figures. Composite figures are shapes that consist of two or more geometric figures.
#### Example Problem
Consider the composite figure shown below:
[Insert Image Here]
The figure is composed of a rectangle and a right triangle. The dimensions are as follows:
- The base of the triangle is 4 units.
- The height of the triangle (which is also the width of the rectangle) is 3 units.
- The length of the rectangle is 10 units.
Let’s break down the steps to calculate the total area of this composite figure.
### Step-by-Step Solution
1. **Find the Area of the Triangle:**
The formula for the area of a triangle is:
\[
\text{Area of a Triangle} = \frac{1}{2} \times \text{base} \times \text{height}
\]
Substitute the given values:
\[
\text{Area of the Triangle} = \frac{1}{2} \times 4 \times 3 = 6 \text{ square units}
\]
2. **Find the Area of the Rectangle:**
The formula for the area of a rectangle is:
\[
\text{Area of a Rectangle} = \text{length} \times \text{width}
\]
Substitute the given values:
\[
\text{Area of the Rectangle} = 10 \times 3 = 30 \text{ square units}
\]
3. **Calculate the Total Area of the Composite Figure:**
Add the areas of the rectangle and the triangle together:
\[
\text{Total Area} = \text{Area of the Triangle} + \text{Area of the Rectangle} = 6 + 30 = 36 \text{ square units}
\]
#### Conclusion
The total area of the composite figure is 36 square units. By breaking down the composite figure into simpler shapes, we can easily calculate the total area. This method can be applied to a variety of composite figures composed of basic geometric shapes.
Transcribed Image Text:## Calculating the Area of Composite Figures: Example
In this example, we will learn how to calculate the area of a composite shape. The shape is an arrow with specific measurements given.
### Diagram Explanation
- The diagram is an arrow shape with segmented parts.
- The entire arrow is composed of a rectangle and a triangular section.
#### Measurements
1. **Rectangle Part**:
- Height: 4 units
- Length: 7 units
2. **Entire Shape**:
- Total Length: 11 units
- Height of Arrowhead: 8 units
#### Important Notes
To calculate the area of this composite shape, follow these steps:
### Step-by-Step Calculation
1. **Calculate the Area of the Rectangle**:
- Formula: Area of Rectangle = Length × Height
- Substituting the values: Area of Rectangle = 7 × 4 = 28 square units
2. **Calculate the Area of the Arrowhead (Triangle)**:
- The height of the triangle is given as 8 units.
- The base of the triangle is the difference between the total length and the length of the rectangle: 11 - 7 = 4 units.
- Formula: Area of Triangle = 1/2 × Base × Height
- Substituting the values: Area of Triangle = 1/2 × 4 × 8 = 16 square units
3. **Calculate the Total Area**:
- Add the areas of the rectangle part and the triangular part to find the total area.
- Total Area = Area of Rectangle + Area of Triangle
- Total Area = 28 + 16 = 44 square units
Therefore, the total area of the composite figure (the arrow) is **44 square units**.
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