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
How would I find the length for an equilateral
![**Question:** What is the length of the side of an equilateral triangle if the height is \( 8\sqrt{3} \)?
**Answer Box:**
Length = _________
**Explanation:**
This question seeks to determine the length of the side of an equilateral triangle, given that the height of the triangle is \( 8\sqrt{3} \).
In an equilateral triangle, all sides are of equal length, and the height can be determined by splitting the triangle into two 30-60-90 right triangles. For a 30-60-90 triangle, the ratio of the sides is 1 : \( \sqrt{3} \) : 2. Thus, the height (the side opposite the 60° angle) is \( \frac{\sqrt{3}}{2} \) times the length of the side of the equilateral triangle.
To solve for the side length \( s \):
\[ h = \frac{\sqrt{3}}{2} s \]
\[ 8\sqrt{3} = \frac{\sqrt{3}}{2} s \]
\[ s = 8\sqrt{3} \times \frac{2}{\sqrt{3}} \]
\[ s = 16 \]
Therefore, the length of the side of the equilateral triangle is 16.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F346b36e0-a4d8-47d7-8bd1-1ad776cb1cf3%2F07cb45f0-ac4e-4443-a162-c330418d3a62%2F3bwgl35_processed.png&w=3840&q=75)
Transcribed Image Text:**Question:** What is the length of the side of an equilateral triangle if the height is \( 8\sqrt{3} \)?
**Answer Box:**
Length = _________
**Explanation:**
This question seeks to determine the length of the side of an equilateral triangle, given that the height of the triangle is \( 8\sqrt{3} \).
In an equilateral triangle, all sides are of equal length, and the height can be determined by splitting the triangle into two 30-60-90 right triangles. For a 30-60-90 triangle, the ratio of the sides is 1 : \( \sqrt{3} \) : 2. Thus, the height (the side opposite the 60° angle) is \( \frac{\sqrt{3}}{2} \) times the length of the side of the equilateral triangle.
To solve for the side length \( s \):
\[ h = \frac{\sqrt{3}}{2} s \]
\[ 8\sqrt{3} = \frac{\sqrt{3}}{2} s \]
\[ s = 8\sqrt{3} \times \frac{2}{\sqrt{3}} \]
\[ s = 16 \]
Therefore, the length of the side of the equilateral triangle is 16.
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