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

Latest revision as of 17:37, 31 March 2023

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

Solution 3

We know the longest side must be denoted by the 4 in the ratio. Since the ratio is 3:3:4, we know that the longest side must be $\frac{4}{3+3+4}$ of the degree total (which for all triangles is 180). Thus, $\frac{4}{3+3+4} \cdot 180 = \frac{4}{10} \cdot 180 = \boxed{\textbf{(D) }72}$

~Ligonmathkid2

Solution 4 (Brute Force) NOT RECOMMENDED

Since we see the ratio is $3:3:4$ , we can rule out the answer of ${\textbf{(E) }90}$ because the numbers in the ratio are too big to have $90^\circ$ . 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 $\boxed{\textbf{(D) }72}$ .

~jason.ca

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

@@ Line 7: / Line 7: @@
 ==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>.
+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==
@@ Line 13: / Line 13: @@
 ~MathFun1000
+==Solution 3==
+We know the longest side must be denoted by the 4 in the ratio. Since the ratio 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>
+~Ligonmathkid2
+==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 (CREATIVE THINKING!!!)==
+https://youtu.be/2CmjcUwuYoE
+~Education, the Study of Everything
 ==Video Solution==

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

Page

Toolbox

Search