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
Related questions
Question
24. Solve for x.
![### Question 24/36
#### Solve for \( x \).
An image is presented showing a right triangle with one of its angles measuring \( 60^\circ \). The hypotenuse of the triangle is labeled as \( 20\sqrt{3} \), and the side opposite to the \( 60^\circ \) angle is labeled as \( x \).
The answer choices provided are:
- A) 60
- B) 20
- C) \( 10\sqrt{3} \)
- D) \( 10\sqrt{2} \)
#### Explanation:
In a right-angled triangle, if one of the angles is \( 60^\circ \), then the triangle is a 30-60-90 triangle. In such triangles, the sides are in the ratio of \( 1 : \sqrt{3} : 2 \).
Given:
- Hypotenuse (\( 20\sqrt{3} \))
- Long leg opposite the \( 60^\circ \) angle (\( x \))
The length of the side opposite the \( 60^\circ \) angle (long leg) is calculated by dividing the hypotenuse by 2 to get the shorter leg and then multiplying it by \( \sqrt{3} \):
\[
x = \frac{20\sqrt{3}}{2} \cdot \sqrt{3} = 10 \cdot 3 = 30
\]
So the correct answer should be 30, but since the provided options don't include 30, it's likely there's an error in either the options or the given data.
Typically, we would expect an answer based on accurate data matching with the \( 30\). We might verify or correct problem constraints based on correctly solved steps.
- Remember, standard 30-60-90 triangles side ratios should clarify calculation results if data context fits correct key choices.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F34bff1e1-af68-4229-b0ef-aa0aaa134e46%2Fdc3201d8-0af5-42eb-8dce-528a2642589e%2F2n4vxb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Question 24/36
#### Solve for \( x \).
An image is presented showing a right triangle with one of its angles measuring \( 60^\circ \). The hypotenuse of the triangle is labeled as \( 20\sqrt{3} \), and the side opposite to the \( 60^\circ \) angle is labeled as \( x \).
The answer choices provided are:
- A) 60
- B) 20
- C) \( 10\sqrt{3} \)
- D) \( 10\sqrt{2} \)
#### Explanation:
In a right-angled triangle, if one of the angles is \( 60^\circ \), then the triangle is a 30-60-90 triangle. In such triangles, the sides are in the ratio of \( 1 : \sqrt{3} : 2 \).
Given:
- Hypotenuse (\( 20\sqrt{3} \))
- Long leg opposite the \( 60^\circ \) angle (\( x \))
The length of the side opposite the \( 60^\circ \) angle (long leg) is calculated by dividing the hypotenuse by 2 to get the shorter leg and then multiplying it by \( \sqrt{3} \):
\[
x = \frac{20\sqrt{3}}{2} \cdot \sqrt{3} = 10 \cdot 3 = 30
\]
So the correct answer should be 30, but since the provided options don't include 30, it's likely there's an error in either the options or the given data.
Typically, we would expect an answer based on accurate data matching with the \( 30\). We might verify or correct problem constraints based on correctly solved steps.
- Remember, standard 30-60-90 triangles side ratios should clarify calculation results if data context fits correct key choices.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 1 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, geometry and related others by exploring similar questions and additional content below.Recommended textbooks for you
Elementary Geometry For College Students, 7e
Geometry
ISBN:
9781337614085
Author:
Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:
Cengage,
Elementary Geometry for College Students
Geometry
ISBN:
9781285195698
Author:
Daniel C. Alexander, Geralyn M. Koeberlein
Publisher:
Cengage Learning
Elementary Geometry For College Students, 7e
Geometry
ISBN:
9781337614085
Author:
Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:
Cengage,
Elementary Geometry for College Students
Geometry
ISBN:
9781285195698
Author:
Daniel C. Alexander, Geralyn M. Koeberlein
Publisher:
Cengage Learning