Given that the angles of a triangle are in the ratio of 2 : 3 : 7 with the length of the smallest side of 2015 units, find the circumradius of the triangle.
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The sum of angles in a triangle is 180.
So, 2k+3k+7k = 180 which implies that 12k = 180 and k=15 degrees.
So, the angles are 30,45 and 105 degrees.
The shortest side is opposite to the shortest angle.
R(circumradius) = abc/4(Area). where a is the shortest side.
Where a,b, and c are side lengths.
Area is also equal to 2 1 b c s i n A .
So, R = 4 2 1 b c s i n A a b c = 2 s i n A a which is popularly known as the sine rule.
Now, R = 2015/1 = 2015 which is the circumradius of the triangle.
Given
The angles of a triangle are in the ratio 2:3:7
⇒ 2x+3x+7x=180
⇒ x=5
Therefore the angles are 30,45,105
We know that
sin A a = sin B b = sin C c =2R
In the above equation, a,b,c are sides of the triangle , R is circumradius
Since given the smallest side is 2015.Therefore the angle opposite to the side 2015 is 30
sin 3 0 2 0 1 5 =2R
⇒ R=2015
Therefore the circumradius of the triangle is 2015
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Angles Ratio = 2:3:7
So 2 x + 3 x + 7 x = 1 8 0 and the angles are 3 0 , 4 5 , 1 0 5
Smallest side is always opposite to smallest angle so
Use a , b , c sides and A , B , C Angles s i n A a = s i n B b = s i n C c = 2 R side_length=2015 units
Solving for Circumradius R s i n 3 0 2 0 1 5 = 2 R Circumradius R = 2 0 1 5 units