Difference between revisions of "2011 AMC 10B Problems/Problem 7"
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| − | == Problem | + | == Problem== |
The sum of two angles of a triangle is <math>\frac{6}{5}</math> of a right angle, and one of these two angles is <math>30^{\circ}</math> larger than the other. What is the degree measure of the largest angle in the triangle? | The sum of two angles of a triangle is <math>\frac{6}{5}</math> of a right angle, and one of these two angles is <math>30^{\circ}</math> larger than the other. What is the degree measure of the largest angle in the triangle? | ||
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== Solution == | == Solution == | ||
| − | The sum of two angles in a triangle is <math>\frac{6}{5}</math> of a right angle <math>\longrightarrow \frac{6}{5} \times | + | The sum of two angles in a triangle is <math>\frac{6}{5}</math> of a right angle <math>\longrightarrow \frac{6}{5} \times 90 = 108</math> |
If <math>x</math> is the measure of the first angle, then the measure of the second angle is <math>x+30</math>. | If <math>x</math> is the measure of the first angle, then the measure of the second angle is <math>x+30</math>. | ||
<cmath>x + x + 30 = 108 \longrightarrow 2x = 78 \longrightarrow x = 39</cmath> | <cmath>x + x + 30 = 108 \longrightarrow 2x = 78 \longrightarrow x = 39</cmath> | ||
| − | Now we know the measure of two angles are <math>39^{\circ}</math> and <math>69^{\circ}</math>. By the Triangle Sum Theorem, the sum of all angles in a triangle is <math>180^{\circ},</math> so the final angle is <math>72^{\circ}</math>. Therefore, the largest angle in the triangle is <math>\boxed{\ | + | Now we know the measure of two angles are <math>39^{\circ}</math> and <math>69^{\circ}</math>. By the Triangle Sum Theorem, the sum of all angles in a triangle is <math>180^{\circ},</math> so the final angle is <math>72^{\circ}</math>. Therefore, the largest angle in the triangle is <math>\boxed{\mathrm{(B) \ } 72}</math> |
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| + | == See Also== | ||
| + | |||
| + | {{AMC10 box|year=2011|ab=B|num-b=6|num-a=8}} | ||
| + | {{MAA Notice}} | ||
Latest revision as of 12:11, 4 July 2013
Problem
The sum of two angles of a triangle is 
of a right angle, and one of these two angles is 
larger than the other. What is the degree measure of the largest angle in the triangle?
Solution
The sum of two angles in a triangle is of a right angle 
If 
is the measure of the first angle, then the measure of the second angle is 
.
![\[x + x + 30 = 108 \longrightarrow 2x = 78 \longrightarrow x = 39\]](http://latex.artofproblemsolving.com/1/8/b/18bf4d3e6772d54562d7ac37e5725731ac30c340.png)
Now we know the measure of two angles are 
and 
. By the Triangle Sum Theorem, the sum of all angles in a triangle is 
so the final angle is 
. Therefore, the largest angle in the triangle is 
See Also
| 2011 AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 6 |
Followed by Problem 8 | |
| 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 AMC 10 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
