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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**The Pythagorean Theorem**
The Pythagorean Theorem is a fundamental relation in Euclidean geometry among the three sides of a right triangle. This theorem is named after the ancient Greek mathematician Pythagoras.
**The Pythagorean Theorem formula:**
\[ a^2 + b^2 = c^2 \]
Where:
- \( a \) = side of the right triangle
- \( b \) = side of the right triangle
- \( c \) = hypotenuse (the longest side of the right triangle)
**Solution:**
Given the right triangle, apply the Pythagorean Theorem to determine the length of the hypotenuse.
**Diagram Explanation:**
There is a right triangle diagram annotated with vertices X, Y, and Z.
- The side \( XY \) is labeled with a length of 6.
- The side \( YZ \) is labeled with a length of 8.
- The side \( XZ \) is labeled with a length of 10.
- Angle \( Y \) is marked as 90°, indicating a right angle.
**Calculation:**
1. \( a = 6 \)
2. \( b = 8 \)
3. Plug the values into the Pythagorean Theorem formula:
\[ 6^2 + 8^2 = c^2 \]
4. Calculate the squares:
\[ 36 + 64 = c^2 \]
5. Sum the squares:
\[ 100 = c^2 \]
6. Take the square root of both sides:
\[ c = \sqrt{100} \]
\[ c = 10 \]
Therefore, the hypotenuse \( c \) is 10 units long, confirming the sides of the triangle fit the Pythagorean Theorem.
**Interactive Practice:**
[Insert interactive problem set here for students to practice applying the Pythagorean Theorem]
---
**Note:** The original image shows a triangle with sides 6, 8, and 10, and correctly demonstrates the use of the Pythagorean Theorem.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9da70a83-7985-4d85-a13b-e9dcca0277bd%2Fb1184c25-bd11-41e2-8121-9c15214942f7%2Fnw85j5_processed.jpeg&w=3840&q=75)
Transcribed Image Text:---
**The Pythagorean Theorem**
The Pythagorean Theorem is a fundamental relation in Euclidean geometry among the three sides of a right triangle. This theorem is named after the ancient Greek mathematician Pythagoras.
**The Pythagorean Theorem formula:**
\[ a^2 + b^2 = c^2 \]
Where:
- \( a \) = side of the right triangle
- \( b \) = side of the right triangle
- \( c \) = hypotenuse (the longest side of the right triangle)
**Solution:**
Given the right triangle, apply the Pythagorean Theorem to determine the length of the hypotenuse.
**Diagram Explanation:**
There is a right triangle diagram annotated with vertices X, Y, and Z.
- The side \( XY \) is labeled with a length of 6.
- The side \( YZ \) is labeled with a length of 8.
- The side \( XZ \) is labeled with a length of 10.
- Angle \( Y \) is marked as 90°, indicating a right angle.
**Calculation:**
1. \( a = 6 \)
2. \( b = 8 \)
3. Plug the values into the Pythagorean Theorem formula:
\[ 6^2 + 8^2 = c^2 \]
4. Calculate the squares:
\[ 36 + 64 = c^2 \]
5. Sum the squares:
\[ 100 = c^2 \]
6. Take the square root of both sides:
\[ c = \sqrt{100} \]
\[ c = 10 \]
Therefore, the hypotenuse \( c \) is 10 units long, confirming the sides of the triangle fit the Pythagorean Theorem.
**Interactive Practice:**
[Insert interactive problem set here for students to practice applying the Pythagorean Theorem]
---
**Note:** The original image shows a triangle with sides 6, 8, and 10, and correctly demonstrates the use of the Pythagorean Theorem.
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