The altitude of the equilateral triangle ARMS is SA = 27. Find the exact length of M.S. S M 27 A R
The altitude of the equilateral triangle ARMS is SA = 27. Find the exact length of M.S. S M 27 A R
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![### Problem Description
The altitude of the equilateral triangle \( \triangle RMS \) is \( SA = 27 \). Find the exact length of \( MS \).
### Diagram Explanation
The given diagram illustrates an equilateral triangle \( \triangle RMS \) with a vertex at \( S \) and the base \( MR \). The altitude \( SA \) extends from vertex \( S \) perpendicular to the base \( MR \), intersecting it at point \( A \). The length of the altitude \( SA \) is specified as 27 units.
### Calculation Steps
To find the length of \( MS \), we can use the properties of equilateral triangles. Given that the altitude of an equilateral triangle divides it into two 30-60-90 triangles, we can use the following properties of 30-60-90 triangles:
1. The length of the side opposite the 30° angle is half the hypotenuse.
2. The length of the side opposite the 60° angle is \( \frac{\sqrt{3}}{2} \) times the hypotenuse.
Given that \( SA \) is the altitude of the equilateral triangle, we use the following relationship for an equilateral triangle with side length \( a \):
\[ \text{Altitude} = \frac{\sqrt{3}}{2} a \]
Here, the altitude \( SA = 27 \). Setting up the equation, we get:
\[ 27 = \frac{\sqrt{3}}{2} a \]
Solving for \( a \):
\[ a = \frac{27 \cdot 2}{\sqrt{3}} \]
\[ a = \frac{54}{\sqrt{3}} \]
Rationalizing the denominator:
\[ a = \frac{54 \sqrt{3}}{3} \]
\[ a = 18 \sqrt{3} \]
Since \( MS \) is a side of the equilateral triangle, the length of \( MS \) is \( 18 \sqrt{3} \).
Thus, the exact length of \( MS \) is:
\[ \boxed{18 \sqrt{3}} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffa42d4ce-de3d-4fed-993c-6008db3043b5%2Fd70d95ac-d51d-4671-bb76-e18e5a6ef70b%2Fnk6v5sc_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Description
The altitude of the equilateral triangle \( \triangle RMS \) is \( SA = 27 \). Find the exact length of \( MS \).
### Diagram Explanation
The given diagram illustrates an equilateral triangle \( \triangle RMS \) with a vertex at \( S \) and the base \( MR \). The altitude \( SA \) extends from vertex \( S \) perpendicular to the base \( MR \), intersecting it at point \( A \). The length of the altitude \( SA \) is specified as 27 units.
### Calculation Steps
To find the length of \( MS \), we can use the properties of equilateral triangles. Given that the altitude of an equilateral triangle divides it into two 30-60-90 triangles, we can use the following properties of 30-60-90 triangles:
1. The length of the side opposite the 30° angle is half the hypotenuse.
2. The length of the side opposite the 60° angle is \( \frac{\sqrt{3}}{2} \) times the hypotenuse.
Given that \( SA \) is the altitude of the equilateral triangle, we use the following relationship for an equilateral triangle with side length \( a \):
\[ \text{Altitude} = \frac{\sqrt{3}}{2} a \]
Here, the altitude \( SA = 27 \). Setting up the equation, we get:
\[ 27 = \frac{\sqrt{3}}{2} a \]
Solving for \( a \):
\[ a = \frac{27 \cdot 2}{\sqrt{3}} \]
\[ a = \frac{54}{\sqrt{3}} \]
Rationalizing the denominator:
\[ a = \frac{54 \sqrt{3}}{3} \]
\[ a = 18 \sqrt{3} \]
Since \( MS \) is a side of the equilateral triangle, the length of \( MS \) is \( 18 \sqrt{3} \).
Thus, the exact length of \( MS \) is:
\[ \boxed{18 \sqrt{3}} \]
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