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

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

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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

@@ Line 5: / Line 5: @@
 <math>\textbf{(A) }18\qquad\textbf{(B) }36\qquad\textbf{(C) }60\qquad\textbf{(D) }72\qquad\textbf{(E) }90</math>
+==Solution 1==
+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==
+We can denote the angles of the triangle as <math>3x</math>, <math>3x</math>, <math>4x</math>. Due to the sum of the angles in a triangle, <math>3x+3x+4x=180^{\circ}\implies x=18^{\circ}</math>. The greatest angle is <math>4x</math> and after substitution we get <math>\boxed{\textbf{(D) }72}</math>.
+~MathFun1000
+==Video Solution (CREATIVE THINKING!!!)==
+https://youtu.be/2CmjcUwuYoE
+~Education, the Study of Everything
 ==Video Solution==
 https://youtu.be/rQUwNC0gqdg?t=635
-==Solution 1==
+https://youtu.be/ykR1ApGP0Qg
-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>.
+~savannahsolver
-==Solution 2==
-We can denote the angles of the triangle as <math>3x</math>, <math>3x</math>, <math>4x</math>. Due to the sum of the angles in a triangle, <math>3x+3x+4x=180^{\circ}\implies x=18^{\circ}</math>. <math>4x</math> is the greatest angle and after substitution we get <math>\boxed{\textbf{(D) }72}</math>.
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