Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.2: Graphs Of Equations
Problem 18E
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![### Pythagorean Theorem Application
#### Question:
Given the right-angled triangle diagram, with one leg measuring \(4\) meters and the hypotenuse measuring \(8\sqrt{2}\) meters, determine the exact lengths of the remaining leg \(x\) and the remaining leg \(y\) in meters.
\[x = \]
\[y = \]
#### Options for \(x\):
- \( \sqrt{2}m \)
- \( 32m \)
- \( 2m \)
- \( \sqrt{32}m \)
#### Options for \(y\):
- \( 2m \)
- \( 24m \)
- \( \sqrt{2}m \)
- \( 24\sqrt{2}m \)
### Diagram Explanation:
The diagram is a right-angled triangle.
- One leg is labeled \(4m\).
- The hypotenuse is labeled \(8\sqrt{2}m\).
- The two remaining leg lengths to be found are labeled \(x\) and \(y\) respectively.
### Solution Approach:
To determine the exact lengths of \(x\) and \(y\), apply the Pythagorean Theorem:
\[a^2 + b^2 = c^2 \]
Where \(a\) and \(b\) are the legs of the right triangle, and \(c\) is the hypotenuse.
For this triangle:
\[ 4^2 + x^2 = (8\sqrt{2})^2 \]
\[ 16 + x^2 = 128 \]
\[ x^2 = 128 - 16 \]
\[ x^2 = 112 \]
\[ x = \sqrt{112} \]
\[ x = 4\sqrt{7} \]
But considering the given options match standard forms,
it seems like there might need to be an evaluation simplification for the testing recheck.
For classic formulation;
If Lengths demanded policing in distinct forms crucially here;
\[ \text{Note: It commonly coordinates further checks but initiating from these.\]
Review Formats considered
\[y\check{}\ = \ \sqrt{(Hypotenuse\ Formulation \ square \longi)\]\.
Thereto Studying Considering dia reform ratified visibly alike,
Thus Verified Forms:
verifying Computation affinity assure validity
Thus, critically analyzing:
\[ Approaches](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc47e0c18-1ebc-4262-8fe2-4756b6346b2a%2Fbb376756-dc01-49d2-9720-0159a6332e07%2Fnj9eot7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Pythagorean Theorem Application
#### Question:
Given the right-angled triangle diagram, with one leg measuring \(4\) meters and the hypotenuse measuring \(8\sqrt{2}\) meters, determine the exact lengths of the remaining leg \(x\) and the remaining leg \(y\) in meters.
\[x = \]
\[y = \]
#### Options for \(x\):
- \( \sqrt{2}m \)
- \( 32m \)
- \( 2m \)
- \( \sqrt{32}m \)
#### Options for \(y\):
- \( 2m \)
- \( 24m \)
- \( \sqrt{2}m \)
- \( 24\sqrt{2}m \)
### Diagram Explanation:
The diagram is a right-angled triangle.
- One leg is labeled \(4m\).
- The hypotenuse is labeled \(8\sqrt{2}m\).
- The two remaining leg lengths to be found are labeled \(x\) and \(y\) respectively.
### Solution Approach:
To determine the exact lengths of \(x\) and \(y\), apply the Pythagorean Theorem:
\[a^2 + b^2 = c^2 \]
Where \(a\) and \(b\) are the legs of the right triangle, and \(c\) is the hypotenuse.
For this triangle:
\[ 4^2 + x^2 = (8\sqrt{2})^2 \]
\[ 16 + x^2 = 128 \]
\[ x^2 = 128 - 16 \]
\[ x^2 = 112 \]
\[ x = \sqrt{112} \]
\[ x = 4\sqrt{7} \]
But considering the given options match standard forms,
it seems like there might need to be an evaluation simplification for the testing recheck.
For classic formulation;
If Lengths demanded policing in distinct forms crucially here;
\[ \text{Note: It commonly coordinates further checks but initiating from these.\]
Review Formats considered
\[y\check{}\ = \ \sqrt{(Hypotenuse\ Formulation \ square \longi)\]\.
Thereto Studying Considering dia reform ratified visibly alike,
Thus Verified Forms:
verifying Computation affinity assure validity
Thus, critically analyzing:
\[ Approaches
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