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
Related questions
Question
![**Problem Statement:**
Find the length of the third side. If necessary, round to the nearest tenth.
**Diagram Explanation:**
The image presents a right triangle with one of the angles marked as a right angle (90 degrees). Two side lengths are given:
- One leg measures 5 units.
- The hypotenuse (the side opposite the right angle) measures 12 units.
**Solution:**
To find the length of the third side (the other leg), we can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (\(c\)) is equal to the sum of the squares of the other two sides (\(a\) and \(b\)):
\[ c^2 = a^2 + b^2 \]
In this problem:
- \(c\) (the hypotenuse) = 12 units
- \(a\) = 5 units
- \(b\) = the length we need to find
Rearranging the formula to solve for \(b\), we get:
\[ b^2 = c^2 - a^2 \]
Substituting the known values:
\[ b^2 = 12^2 - 5^2 \]
\[ b^2 = 144 - 25 \]
\[ b^2 = 119 \]
Finally, take the square root of both sides to solve for \(b\):
\[ b = \sqrt{119} \]
\[ b \approx 10.9 \]
Thus, the length of the third side is approximately 10.9 units.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F08807450-21ed-4e55-84b1-e841f0af059c%2F07253c3e-1569-409a-8c68-7a8048388af0%2Frsh8juv_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the length of the third side. If necessary, round to the nearest tenth.
**Diagram Explanation:**
The image presents a right triangle with one of the angles marked as a right angle (90 degrees). Two side lengths are given:
- One leg measures 5 units.
- The hypotenuse (the side opposite the right angle) measures 12 units.
**Solution:**
To find the length of the third side (the other leg), we can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (\(c\)) is equal to the sum of the squares of the other two sides (\(a\) and \(b\)):
\[ c^2 = a^2 + b^2 \]
In this problem:
- \(c\) (the hypotenuse) = 12 units
- \(a\) = 5 units
- \(b\) = the length we need to find
Rearranging the formula to solve for \(b\), we get:
\[ b^2 = c^2 - a^2 \]
Substituting the known values:
\[ b^2 = 12^2 - 5^2 \]
\[ b^2 = 144 - 25 \]
\[ b^2 = 119 \]
Finally, take the square root of both sides to solve for \(b\):
\[ b = \sqrt{119} \]
\[ b \approx 10.9 \]
Thus, the length of the third side is approximately 10.9 units.
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