Use Heron's formula to find the area of the triangle with side lengths 7, 13, and 14, as shown below. Round your answer to the nearest tenth and do not include units in your answer.
Use Heron's formula to find the area of the triangle with side lengths 7, 13, and 14, as shown below. Round your answer to the nearest tenth and do not include units in your answer.
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE:
1. Give the measures of the complement and the supplement of an angle measuring 35°.
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Question
![**Question**
Use Heron’s formula to find the area of the triangle with side lengths 7, 13, and 14, as shown below.
Round your answer to the nearest tenth and do not include units in your answer.
[Diagram: A triangle with sides labeled 7, 13, and 14]
**Explanation of Heron’s Formula**
Heron's formula provides a way to calculate the area of a triangle when you know the lengths of all three sides. The formula is:
\[ \text{Area} = \sqrt{s(s-a)(s-b)(s-c)} \]
where \( a \), \( b \), and \( c \) are the side lengths of the triangle, and \( s \) is the semi-perimeter of the triangle, calculated as:
\[ s = \frac{a+b+c}{2} \]
**Steps to Solve:**
1. Calculate the semi-perimeter:
\[ s = \frac{7 + 13 + 14}{2} = 17 \]
2. Use Heron’s formula to calculate the area:
\[ \text{Area} = \sqrt{17(17-7)(17-13)(17-14)} \]
\[ = \sqrt{17 \times 10 \times 4 \times 3} \]
\[ = \sqrt{2040} \]
\[ \approx 45.1 \]
**Final Answer:**
The area of the triangle is approximately 45.1.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F25e500be-5feb-4687-804d-d701ad7786f6%2Fef96f8f8-d482-4fb1-acb0-e3dbd9354caa%2Fjerdf6_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Question**
Use Heron’s formula to find the area of the triangle with side lengths 7, 13, and 14, as shown below.
Round your answer to the nearest tenth and do not include units in your answer.
[Diagram: A triangle with sides labeled 7, 13, and 14]
**Explanation of Heron’s Formula**
Heron's formula provides a way to calculate the area of a triangle when you know the lengths of all three sides. The formula is:
\[ \text{Area} = \sqrt{s(s-a)(s-b)(s-c)} \]
where \( a \), \( b \), and \( c \) are the side lengths of the triangle, and \( s \) is the semi-perimeter of the triangle, calculated as:
\[ s = \frac{a+b+c}{2} \]
**Steps to Solve:**
1. Calculate the semi-perimeter:
\[ s = \frac{7 + 13 + 14}{2} = 17 \]
2. Use Heron’s formula to calculate the area:
\[ \text{Area} = \sqrt{17(17-7)(17-13)(17-14)} \]
\[ = \sqrt{17 \times 10 \times 4 \times 3} \]
\[ = \sqrt{2040} \]
\[ \approx 45.1 \]
**Final Answer:**
The area of the triangle is approximately 45.1.
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