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
![### Finding the Height of a Triangle
#### Problem Statement:
b. Find the height \( h \) of the triangle.
#### Diagram Explanation:
The given image shows a right-angled triangle with the following side lengths:
- The base of the triangle is 8 cm.
- One side of the triangle (not the hypotenuse) is 4 cm.
- The hypotenuse (the longest side opposite the right angle) is 10 cm.
In the diagram, the height \( h \) is perpendicular to the given base and forms a right angle with it. The height splits the original triangle into two smaller right-angled triangles.
#### Solution:
To find the height \( h \), we can use the Pythagorean theorem, which states:
\[ a^2 + b^2 = c^2 \]
Here, we can consider the two right-angled triangles formed. Let's solve it step by step:
1. **Identify the right triangles and apply the Pythagorean Theorem**:
- For the smaller triangle with sides \( h \) and 4 cm, and hypotenuse 10 cm:
\[ h^2 + 4^2 = 10^2 \]
\[ h^2 + 16 = 100 \]
\[ h^2 = 84 \]
\[ h = \sqrt{84} \]
\[ h = 2\sqrt{21} \, \text{cm} \]
2. **Verifying the height**:
- Ensure the obtained height is reasonable considering the overall dimensions of the original triangle.
#### Conclusion:
The height \( h \) of the triangle is \( 2\sqrt{21} \, \text{cm} \).
Understanding the geometric relations in triangles is crucial for solving problems involving heights. The Pythagorean theorem provides a reliable method when dealing with right-angled triangles.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff8311114-a7cb-49e3-b44e-1e05de4bc29a%2Fac0c87ae-8f61-449c-87f3-694dd096a805%2Fob2cmoo.png&w=3840&q=75)
Transcribed Image Text:### Finding the Height of a Triangle
#### Problem Statement:
b. Find the height \( h \) of the triangle.
#### Diagram Explanation:
The given image shows a right-angled triangle with the following side lengths:
- The base of the triangle is 8 cm.
- One side of the triangle (not the hypotenuse) is 4 cm.
- The hypotenuse (the longest side opposite the right angle) is 10 cm.
In the diagram, the height \( h \) is perpendicular to the given base and forms a right angle with it. The height splits the original triangle into two smaller right-angled triangles.
#### Solution:
To find the height \( h \), we can use the Pythagorean theorem, which states:
\[ a^2 + b^2 = c^2 \]
Here, we can consider the two right-angled triangles formed. Let's solve it step by step:
1. **Identify the right triangles and apply the Pythagorean Theorem**:
- For the smaller triangle with sides \( h \) and 4 cm, and hypotenuse 10 cm:
\[ h^2 + 4^2 = 10^2 \]
\[ h^2 + 16 = 100 \]
\[ h^2 = 84 \]
\[ h = \sqrt{84} \]
\[ h = 2\sqrt{21} \, \text{cm} \]
2. **Verifying the height**:
- Ensure the obtained height is reasonable considering the overall dimensions of the original triangle.
#### Conclusion:
The height \( h \) of the triangle is \( 2\sqrt{21} \, \text{cm} \).
Understanding the geometric relations in triangles is crucial for solving problems involving heights. The Pythagorean theorem provides a reliable method when dealing with right-angled triangles.
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