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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![### Problem: Solving for x and y in a Right Triangle
#### Question:
Solve for \( x \) and \( y \). Select BOTH correct answers.
#### Diagram:
- The diagram shows a right triangle with one angle measuring \( 30^\circ \).
- The hypotenuse of the triangle is labeled \( 14 \).
- The side opposite the \( 30^\circ \) angle is labeled \( y \).
- The side adjacent to the \( 30^\circ \) angle and opposite the right angle is labeled \( x \).
#### Answer Choices:
- [ ] \( x = 7\sqrt{3} \)
- [ ] \( x = 7 \)
- [ ] \( y = 7\sqrt{3} \)
- [ ] \( y = 7 \)
**Explanation:**
In a right triangle with angles \( 30^\circ, 60^\circ\), and \( 90^\circ \):
- The length of the side opposite the \( 30^\circ \) angle is half the length of the hypotenuse.
- The length of the side opposite the \( 60^\circ \) angle (adjacent to the \( 30^\circ \) angle) is \( \sqrt{3} \) times the length of the side opposite the \( 30^\circ \) angle.
Using these properties:
- Given hypotenuse \( = 14 \)
- \( y = \frac{1}{2} \times 14 = 7 \)
- \( x = 7\sqrt{3} \)
Thus, the correct answers are:
- \( x = 7\sqrt{3} \)
- \( y = 7 \)
Ensure to select both correct answers to solve for \( x \) and \( y \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8013d441-5946-4d9d-b85b-992de95a8487%2F854da6fa-4894-493c-9490-e42d2579591d%2F9op46sq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem: Solving for x and y in a Right Triangle
#### Question:
Solve for \( x \) and \( y \). Select BOTH correct answers.
#### Diagram:
- The diagram shows a right triangle with one angle measuring \( 30^\circ \).
- The hypotenuse of the triangle is labeled \( 14 \).
- The side opposite the \( 30^\circ \) angle is labeled \( y \).
- The side adjacent to the \( 30^\circ \) angle and opposite the right angle is labeled \( x \).
#### Answer Choices:
- [ ] \( x = 7\sqrt{3} \)
- [ ] \( x = 7 \)
- [ ] \( y = 7\sqrt{3} \)
- [ ] \( y = 7 \)
**Explanation:**
In a right triangle with angles \( 30^\circ, 60^\circ\), and \( 90^\circ \):
- The length of the side opposite the \( 30^\circ \) angle is half the length of the hypotenuse.
- The length of the side opposite the \( 60^\circ \) angle (adjacent to the \( 30^\circ \) angle) is \( \sqrt{3} \) times the length of the side opposite the \( 30^\circ \) angle.
Using these properties:
- Given hypotenuse \( = 14 \)
- \( y = \frac{1}{2} \times 14 = 7 \)
- \( x = 7\sqrt{3} \)
Thus, the correct answers are:
- \( x = 7\sqrt{3} \)
- \( y = 7 \)
Ensure to select both correct answers to solve for \( x \) and \( y \).
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