*13, Given that the rectangle ABCD in Figure 12.41 has area 108 square units, determine the area of the shaded triangle. Explain your reasoning. D A 101 12 units 6 units C B Figure 12.41 Determining the area of the shaded triangle. T not alumiol
*13, Given that the rectangle ABCD in Figure 12.41 has area 108 square units, determine the area of the shaded triangle. Explain your reasoning. D A 101 12 units 6 units C B Figure 12.41 Determining the area of the shaded triangle. T not alumiol
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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Question
Explain your reasoning.
![### Problem 13: Determining the Area of a Shaded Triangle
#### Given Information:
The rectangle \(ABCD\) in **Figure 12.41** has an area of \(108\) square units. You are required to determine the area of the shaded triangle within the rectangle. Please explain your reasoning.
#### Diagram Analysis:
The diagram features rectangle \(ABCD\) where:
- \(AB\) is the base with a length of \(12\) units.
- \(AD\) is the height with a length of \(6\) units.
- There is a shaded triangle inside the rectangle, forming from points \(B\), \(C\), and a point on line segment \(AD\).
#### Explanation:
1. **Area of Rectangle \(ABCD\):**
Given the area of the rectangle is \(108\) square units.
\[
\text{Area of Rectangle} = \text{base} \times \text{height}
\]
With the base \(AB = 12\) units, let’s denote the height \(AD\) as \(h\) (although \(h\) is given to be \(6\) units).
\[
12 \times 6 = 108 \ \text{square units}
\]
Hence, the dimensions given align perfectly with the provided area.
2. **Area of Shaded Triangle:**
This is a right triangle where the base \(BC\) and the height \(AB\) are:
\[
\text{base} = 12 \ \text{units}
\]
\[
\text{height} = 6 \ \text{units}
\]
The formula for the area of a triangle is:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]
Substituting the known values:
\[
\text{Area} = \frac{1}{2} \times 12 \times 6
\]
\[
\text{Area} = \frac{1}{2} \times 72
\]
\[
\text{Area} = 36 \ \text{square units}
\]
Therefore, the area of the shaded triangle is **36 square units](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7c60e9dc-9541-469d-8098-00c54f69ea68%2F91b02ddb-652f-46ee-9c07-4af2ee01a6d1%2Flplrp9i_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem 13: Determining the Area of a Shaded Triangle
#### Given Information:
The rectangle \(ABCD\) in **Figure 12.41** has an area of \(108\) square units. You are required to determine the area of the shaded triangle within the rectangle. Please explain your reasoning.
#### Diagram Analysis:
The diagram features rectangle \(ABCD\) where:
- \(AB\) is the base with a length of \(12\) units.
- \(AD\) is the height with a length of \(6\) units.
- There is a shaded triangle inside the rectangle, forming from points \(B\), \(C\), and a point on line segment \(AD\).
#### Explanation:
1. **Area of Rectangle \(ABCD\):**
Given the area of the rectangle is \(108\) square units.
\[
\text{Area of Rectangle} = \text{base} \times \text{height}
\]
With the base \(AB = 12\) units, let’s denote the height \(AD\) as \(h\) (although \(h\) is given to be \(6\) units).
\[
12 \times 6 = 108 \ \text{square units}
\]
Hence, the dimensions given align perfectly with the provided area.
2. **Area of Shaded Triangle:**
This is a right triangle where the base \(BC\) and the height \(AB\) are:
\[
\text{base} = 12 \ \text{units}
\]
\[
\text{height} = 6 \ \text{units}
\]
The formula for the area of a triangle is:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]
Substituting the known values:
\[
\text{Area} = \frac{1}{2} \times 12 \times 6
\]
\[
\text{Area} = \frac{1}{2} \times 72
\]
\[
\text{Area} = 36 \ \text{square units}
\]
Therefore, the area of the shaded triangle is **36 square units
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