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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![**Directions:** Solve for the variables \(x\), \(y\) and \(z\) in each triangle:
*Diagram Explanation:*
This triangular diagram shows a larger right triangle \( \triangle GHK \) with vertex \( G \), \( H \), and \( K \). The right angle is at vertex \( H \). There are two smaller right triangles within this larger triangle:
1. \( \triangle GHK \):
- \( GH = z \)
- \( GK = 9 \)
- \( KH = y \)
2. \( \triangle HIK \) (inside \( \triangle GHK \)):
- \( HI = 3 \)
- \( IK = x \)
- \( \angle IHK \) is a right angle
Given the right angle interior to both triangles, you can apply the Pythagorean theorem to find \( x \), \( y \), and \( z \).
**Answer:**
To find the values of \( x \), \( y \), and \( z \):
1. **In \( \triangle HIK \)**:
Using the Pythagorean theorem:
\[ x^2 + 3^2 = y^2 \]
\[ x^2 + 9 = y^2 \]
\[ y^2 - 9 = x^2 \]
\[ y = \sqrt{x^2 + 9} \]
2. **In \( \triangle GHK \)**:
Using the Pythagorean theorem:
\[ GH^2 + KH^2 = GK^2 \]
\[ z^2 + y^2 = 9^2 \]
\[ z^2 + y^2 = 81 \]
Substitute \( y \) with \( \sqrt{x^2 + 9} \):
\[ z^2 + (\sqrt{x^2 + 9})^2 = 81 \]
\[ z^2 + x^2 + 9 = 81 \]
\[ z^2 + x^2 = 72 \]
\[ z = \sqrt{72 - x^2} \]
So, the values of \( x \), \( y \), and \( z \) in terms of solving a specific value or multiple values can be obtained from the above expressions:](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F87804e42-439c-4b15-9fec-573b7a02348d%2Fc4581fdf-1144-4f2d-99a1-974dee057034%2Ftmgk3i2_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Directions:** Solve for the variables \(x\), \(y\) and \(z\) in each triangle:
*Diagram Explanation:*
This triangular diagram shows a larger right triangle \( \triangle GHK \) with vertex \( G \), \( H \), and \( K \). The right angle is at vertex \( H \). There are two smaller right triangles within this larger triangle:
1. \( \triangle GHK \):
- \( GH = z \)
- \( GK = 9 \)
- \( KH = y \)
2. \( \triangle HIK \) (inside \( \triangle GHK \)):
- \( HI = 3 \)
- \( IK = x \)
- \( \angle IHK \) is a right angle
Given the right angle interior to both triangles, you can apply the Pythagorean theorem to find \( x \), \( y \), and \( z \).
**Answer:**
To find the values of \( x \), \( y \), and \( z \):
1. **In \( \triangle HIK \)**:
Using the Pythagorean theorem:
\[ x^2 + 3^2 = y^2 \]
\[ x^2 + 9 = y^2 \]
\[ y^2 - 9 = x^2 \]
\[ y = \sqrt{x^2 + 9} \]
2. **In \( \triangle GHK \)**:
Using the Pythagorean theorem:
\[ GH^2 + KH^2 = GK^2 \]
\[ z^2 + y^2 = 9^2 \]
\[ z^2 + y^2 = 81 \]
Substitute \( y \) with \( \sqrt{x^2 + 9} \):
\[ z^2 + (\sqrt{x^2 + 9})^2 = 81 \]
\[ z^2 + x^2 + 9 = 81 \]
\[ z^2 + x^2 = 72 \]
\[ z = \sqrt{72 - x^2} \]
So, the values of \( x \), \( y \), and \( z \) in terms of solving a specific value or multiple values can be obtained from the above expressions:
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