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Difference between revisions of "2017 AMC 8 Problems/Problem 6"

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==Solution 1==
 
==Solution 1==
  
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The sum of the ratios is <math>10</math>.  Since the sum of the angles of a triangle is <math>180^{\circ}</math>, the ratio can be scaled up to <math>54:54:72</math> <math>(3\cdot 18:3\cdot 18:4\cdot 18)</math>. The numbers in the ratio <math>54:54:72</math> represent the angles of the triangle.  The question asks for the largest, so the answer is <math>\boxed{\textbf{(D) }72}</math>.
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The sum of the ratios is <math>10</math>.  Since the sum of the angles of a triangle is <math>180^{\circ}</math>, the ratio can be scaled up to <math>54:54:72</math> <math>(3\cdot 18:3\cdot 18:4\cdot 18).</math>  The numbers in the ratio <math>54:54:72</math> represent the angles of the triangle.  The question asks for the largest, so the answer is <math>\boxed{\textbf{(D) }72}</math>.
  
 
==Solution 2==
 
==Solution 2==
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~MathFun1000
 
~MathFun1000
  
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==Solution 3 (Brute Force) NOT RECOMMENDED==
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==Video Solution (CREATIVE THINKING!!!)==
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https://youtu.be/2CmjcUwuYoE
  
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Since we see the ratio is <math>3:3:4</math>, we can rule out the answer of <math>{\textbf{(E) }90}</math> because the numbers in the ratio are to big to have <math>90^\circ</math>. Also, we are trying to find the largest angle and all the other angles except for 72 are to small to be the largest angle. Using all this, our answer is <math>{\textbf{(D) }72}</math>.
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~Education, the Study of Everything
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~jason.ca
 
  
 
==Video Solution==
 
==Video Solution==

Latest revision as of 14:27, 26 May 2024

Problem

If the degree measures of the angles of a triangle are in the ratio $3:3:4$, what is the degree measure of the largest angle of the triangle?

$\textbf{(A) }18\qquad\textbf{(B) }36\qquad\textbf{(C) }60\qquad\textbf{(D) }72\qquad\textbf{(E) }90$

Solution 1

The sum of the ratios is $10$. Since the sum of the angles of a triangle is $180^{\circ}$, the ratio can be scaled up to $54:54:72$ $(3\cdot 18:3\cdot 18:4\cdot 18).$ The numbers in the ratio $54:54:72$ represent the angles of the triangle. The question asks for the largest, so the answer is $\boxed{\textbf{(D) }72}$.

Solution 2

We can denote the angles of the triangle as $3x$, $3x$, $4x$. Due to the sum of the angles in a triangle, $3x+3x+4x=180^{\circ}\implies x=18^{\circ}$. The greatest angle is $4x$ and after substitution we get $\boxed{\textbf{(D) }72}$.

~MathFun1000

Video Solution (CREATIVE THINKING!!!)

https://youtu.be/2CmjcUwuYoE

~Education, the Study of Everything

Video Solution

https://youtu.be/rQUwNC0gqdg?t=635

https://youtu.be/ykR1ApGP0Qg

~savannahsolver

See Also

2017 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 5 		Followed by
Problem 7
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AJHSME/AMC 8 Problems and Solutions

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