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 14: / Line 14: @@
 ~MathFun1000
-==Solution 3==
+==Video Solution (CREATIVE THINKING!!!)==
-We know the longest side must be denoted by the 4 in the ratio. Since the ration is 3:3:4, we know that the longest side must be <math>\frac{4}{3+3+4}</math> of the degree total (which for all triangles is 180). Thus, <cmath>\frac{4}{3+3+4} \cdot 180 = \frac{4}{10} \cdot 180 = \boxed{\textbf{(D) }72}</cmath>
+https://youtu.be/2CmjcUwuYoE
-~Ligonmathkid2
+~Education, the Study of Everything
-==Solution 4 (Brute Force) NOT RECOMMENDED==
-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 too 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 too small to be the largest angle. Using all this, our answer is <math>\boxed{\textbf{(D) }72}</math>.
-~jason.ca
 ==Video Solution==

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