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
![**Pythagorean Theorem (Level 2)**
**Problem Statement:**
Find the length of the third side. If necessary, write in simplest radical form.
**Diagram Explanation:**
A right triangle is depicted on graph paper. One leg of the triangle is labeled with a length of \(8\), and the hypotenuse (the side opposite the right angle) is labeled with a length of \(2\sqrt{41}\).
**Solution:**
Using the Pythagorean Theorem, which states \(a^2 + b^2 = c^2\), where \(a\) and \(b\) are the legs of the triangle and \(c\) is the hypotenuse, we can find the length of the missing side.
Given:
- One leg (\(a\)) = \(8\)
- Hypotenuse (\(c\)) = \(2\sqrt{41}\)
The Pythagorean Theorem provides the relationship:
\[ a^2 + b^2 = c^2 \]
Plugging in the known values:
\[ 8^2 + b^2 = (2\sqrt{41})^2 \]
\[ 64 + b^2 = 4 \cdot 41 \]
\[ 64 + b^2 = 164 \]
Solving for \(b^2\):
\[ b^2 = 164 - 64 \]
\[ b^2 = 100 \]
\[ b = \sqrt{100} \]
\[ b = 10 \]
Therefore, the length of the missing side is 10.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F79629784-9084-461f-9ae9-4cf541f3f365%2Fff2a5c54-88aa-4a1b-b4e3-2d0a9b243ca0%2Fva2rhr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Pythagorean Theorem (Level 2)**
**Problem Statement:**
Find the length of the third side. If necessary, write in simplest radical form.
**Diagram Explanation:**
A right triangle is depicted on graph paper. One leg of the triangle is labeled with a length of \(8\), and the hypotenuse (the side opposite the right angle) is labeled with a length of \(2\sqrt{41}\).
**Solution:**
Using the Pythagorean Theorem, which states \(a^2 + b^2 = c^2\), where \(a\) and \(b\) are the legs of the triangle and \(c\) is the hypotenuse, we can find the length of the missing side.
Given:
- One leg (\(a\)) = \(8\)
- Hypotenuse (\(c\)) = \(2\sqrt{41}\)
The Pythagorean Theorem provides the relationship:
\[ a^2 + b^2 = c^2 \]
Plugging in the known values:
\[ 8^2 + b^2 = (2\sqrt{41})^2 \]
\[ 64 + b^2 = 4 \cdot 41 \]
\[ 64 + b^2 = 164 \]
Solving for \(b^2\):
\[ b^2 = 164 - 64 \]
\[ b^2 = 100 \]
\[ b = \sqrt{100} \]
\[ b = 10 \]
Therefore, the length of the missing side is 10.
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