Difference between revisions of "2017 AMC 8 Problems/Problem 6"
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| − | ==Problem | + | ==Problem== |
If the degree measures of the angles of a triangle are in the ratio <math>3:3:4</math>, what is the degree measure of the largest angle of the triangle? | If the degree measures of the angles of a triangle are in the ratio <math>3:3:4</math>, what is the degree measure of the largest angle of the triangle? | ||
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<math>\textbf{(A) }18\qquad\textbf{(B) }36\qquad\textbf{(C) }60\qquad\textbf{(D) }72\qquad\textbf{(E) }90</math> | <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 | ||
| + | |||
| + | https://youtu.be/ykR1ApGP0Qg | ||
| + | |||
| + | ~savannahsolver | ||
| + | |||
| + | ==See Also== | ||
| + | {{AMC8 box|year=2017|num-b=5|num-a=7}} | ||
| + | |||
| + | {{MAA Notice}} | ||
Latest revision as of 14:27, 26 May 2024
Contents
Problem
If the degree measures of the angles of a triangle are in the ratio 
, what is the degree measure of the largest angle of the triangle?
Solution 1
The sum of the ratios is 
. Since the sum of the angles of a triangle is 
, the ratio can be scaled up to 
The numbers in the ratio represent the angles of the triangle. The question asks for the largest, so the answer is 
.
Solution 2
We can denote the angles of the triangle as 
, , 
. Due to the sum of the angles in a triangle, 
. The greatest angle is and after substitution we get .
~MathFun1000
Video Solution (CREATIVE THINKING!!!)
~Education, the Study of Everything
Video Solution
https://youtu.be/rQUwNC0gqdg?t=635
~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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
