14) Find the area of the triangle below... 37 (37-20) (84-24) (37-3) +37·17·1317 20 in 557239 239,1 A. 478.2 in² B. 298.9 in² 85° 30 in 24 in C239.1 in² 358.6 in²
14) Find the area of the triangle below... 37 (37-20) (84-24) (37-3) +37·17·1317 20 in 557239 239,1 A. 478.2 in² B. 298.9 in² 85° 30 in 24 in C239.1 in² 358.6 in²
Holt Mcdougal Larson Pre-algebra: Student Edition 2012
1st Edition
ISBN:9780547587776
Author:HOLT MCDOUGAL
Publisher:HOLT MCDOUGAL
Chapter10: Measurement, Area, And Volume
Section10.4: Circumference And Area Of A Circle
Problem 33E
Related questions
Question
![**Area Calculation of a Triangle with Given Dimensions and Angles**
### Problem Statement:
**(14)** Find the area of the triangle below:
### Diagram and Given Information:
A triangle with sides labeled as follows:
- One side = 30 inches
- Another side = 24 inches
- Third side = 20 inches
The angle opposite the 20 inch side is 85 degrees.
### Multiple Choice Answers:
A. 4782.2 in²
B. 298.9 in²
C. 239.1 in²
D. 358.6 in²
### Step-by-Step Solution:
1. **Law of Cosines to Find Third Side:**
The triangle's given angle of 85° allows for using the Law of Cosines:
\(c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\)
Here, \(a = 30\) inches, \(b = 24\) inches, and \(C = 85^\circ\):
\[
c^2 = 30^2 + 24^2 - 2 \cdot 30 \cdot 24 \cdot \cos(85^\circ)
\]
Solve for \(c\) to find the unknown side's length.
2. **Using Known Sides and Angles to Calculate Area:**
The area (\(\Delta\)) of a triangle given two sides and the included angle can be found using the formula:
\[
\Delta = \frac{1}{2}ab \sin(C)
\]
Substitute \(a\), \(b\), and \(C\) into the formula:
\[
\Delta = \frac{1}{2} \cdot 30 \cdot 24 \cdot \sin(85^\circ)
\]
3. **Calculate:**
Approximate \(\sin(85^\circ) \approx 0.9962\):
\[
\Delta = \frac{1}{2} \cdot 30 \cdot 24 \cdot 0.9962 \approx 239.1 \text{ in}^2
\]
**Answer:**
The correct answer is:
C. 239.1 in²
### Explanation of Handwritten Notes:](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa5a061b4-0108-4a51-81fc-44207999c893%2F224cc5d5-cf2c-4bac-b52e-55b9997b1225%2F97yvqic_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Area Calculation of a Triangle with Given Dimensions and Angles**
### Problem Statement:
**(14)** Find the area of the triangle below:
### Diagram and Given Information:
A triangle with sides labeled as follows:
- One side = 30 inches
- Another side = 24 inches
- Third side = 20 inches
The angle opposite the 20 inch side is 85 degrees.
### Multiple Choice Answers:
A. 4782.2 in²
B. 298.9 in²
C. 239.1 in²
D. 358.6 in²
### Step-by-Step Solution:
1. **Law of Cosines to Find Third Side:**
The triangle's given angle of 85° allows for using the Law of Cosines:
\(c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\)
Here, \(a = 30\) inches, \(b = 24\) inches, and \(C = 85^\circ\):
\[
c^2 = 30^2 + 24^2 - 2 \cdot 30 \cdot 24 \cdot \cos(85^\circ)
\]
Solve for \(c\) to find the unknown side's length.
2. **Using Known Sides and Angles to Calculate Area:**
The area (\(\Delta\)) of a triangle given two sides and the included angle can be found using the formula:
\[
\Delta = \frac{1}{2}ab \sin(C)
\]
Substitute \(a\), \(b\), and \(C\) into the formula:
\[
\Delta = \frac{1}{2} \cdot 30 \cdot 24 \cdot \sin(85^\circ)
\]
3. **Calculate:**
Approximate \(\sin(85^\circ) \approx 0.9962\):
\[
\Delta = \frac{1}{2} \cdot 30 \cdot 24 \cdot 0.9962 \approx 239.1 \text{ in}^2
\]
**Answer:**
The correct answer is:
C. 239.1 in²
### Explanation of Handwritten Notes:
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