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CBSE Class 10 Answered

Triangle has perimeter of 6 + 2√3 . One of the angles in the triangle is equal to the exterior angle of a regular hexagon another angle is equal to the exterior angle of a regular 12-sided polygon. Find area of the triangle.
Asked by rushabh1234 | 07 Feb, 2019, 10:46: AM
answered-by-expert Expert Answer
Interior angle of regular polygon of n sides =  [ (n-2)/n ] π
 
Interior angle of regular hexagon = (2/3)π
 
Hence if one of the angle of triangle is equal to exterior angle of regular hexagon, then this angle = π/3 = 60°
 
Interior angle of regular polygon of 12 sides =  (5/6)π
 
Hence if one of the angle of triangle is equal to exterior angle of regular polygon of 12 sides,
then this angle = π/6 = 30°
 
hence angles of triangle are 30°, 60° and 90°.
 
we know that if a triangle has angles 30°, 60° and 90° , then the sides are in the ratio 1 : √3 : 2
 
Let x be the side opposite to 30° angle, then (√3)x be the side opposite to 60° and 2x be the side opposite to right angles.
 
Hence perimeter,  x + (√3)x + 2x = 6+2√3   or x = 2.
 
hence sides of the right angles are 2 and 2√3  . Hence area = (1/2)×2×2√3 = 2√3 sq. units
Answered by Thiyagarajan K | 07 Feb, 2019, 03:27: PM

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