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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Will you solve #6 and #8. Please determine the value of the
![## Understanding Right-Angled Triangles
### Diagram 1:
- This is a right-angled triangle.
- The lengths of the sides are labeled as follows:
- The base (adjacent side to angle θ) is 6 units.
- The height (opposite side to angle θ) is 9 units.
- The hypotenuse (the longest side opposite the right angle) is 15 units.
- Angle θ is one of the non-right angles in the triangle.
### Diagram 2:
- This is also a right-angled triangle.
- The lengths of the sides are labeled as follows:
- The base (adjacent side to angle β) is 4 units.
- The height (opposite side to angle β) is 5 units.
- The hypotenuse (the longest side opposite the right angle) is 6 units.
- Angle β is one of the non-right angles in the triangle.
### Explanation
In a right-angled triangle, the Pythagorean theorem holds true, which can be stated as:
\[ a^2 + b^2 = c^2 \]
where \( a \) and \( b \) are the lengths of the legs of the triangle, and \( c \) is the length of the hypotenuse.
#### Applying the Pythagorean Theorem:
For Diagram 1:
\[ 6^2 + 9^2 = 15^2 \]
\[ 36 + 81 = 225 \]
\[ 117 = 225 \]
(Note: There seems to be an inconsistency in the given side lengths and they do not satisfy the Pythagorean theorem)
For Diagram 2:
\[ 4^2 + 5^2 = 6^2 \]
\[ 16 + 25 = 36 \]
\[ 41 \ne 36 \]
(Note: There seems to be an inconsistency in the given side lengths and they do not satisfy the Pythagorean theorem)
### Important Note:
The side lengths given in both triangles do not satisfy the Pythagorean theorem. Double-check the measurements or verify the provided data for accurate calculations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2eab4935-c027-4811-a454-4c5a0231737d%2Ffbd4f45b-32ef-4d94-a368-d9607e6da744%2F9j1bv5r.jpeg&w=3840&q=75)
Transcribed Image Text:## Understanding Right-Angled Triangles
### Diagram 1:
- This is a right-angled triangle.
- The lengths of the sides are labeled as follows:
- The base (adjacent side to angle θ) is 6 units.
- The height (opposite side to angle θ) is 9 units.
- The hypotenuse (the longest side opposite the right angle) is 15 units.
- Angle θ is one of the non-right angles in the triangle.
### Diagram 2:
- This is also a right-angled triangle.
- The lengths of the sides are labeled as follows:
- The base (adjacent side to angle β) is 4 units.
- The height (opposite side to angle β) is 5 units.
- The hypotenuse (the longest side opposite the right angle) is 6 units.
- Angle β is one of the non-right angles in the triangle.
### Explanation
In a right-angled triangle, the Pythagorean theorem holds true, which can be stated as:
\[ a^2 + b^2 = c^2 \]
where \( a \) and \( b \) are the lengths of the legs of the triangle, and \( c \) is the length of the hypotenuse.
#### Applying the Pythagorean Theorem:
For Diagram 1:
\[ 6^2 + 9^2 = 15^2 \]
\[ 36 + 81 = 225 \]
\[ 117 = 225 \]
(Note: There seems to be an inconsistency in the given side lengths and they do not satisfy the Pythagorean theorem)
For Diagram 2:
\[ 4^2 + 5^2 = 6^2 \]
\[ 16 + 25 = 36 \]
\[ 41 \ne 36 \]
(Note: There seems to be an inconsistency in the given side lengths and they do not satisfy the Pythagorean theorem)
### Important Note:
The side lengths given in both triangles do not satisfy the Pythagorean theorem. Double-check the measurements or verify the provided data for accurate calculations.
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