KVPY2019MathematicsProperties of Triangles
Let l > 0 be a real number, C denote a circle with circumference l and T denote a triangle with perimeter l . Then
Options
- Agiven any positive real number α , we can choose C and T as above such that ratio  Area  ( C )
- Bgiven any positive real number α , we can choose C and T as above such that ratio  Area  ( C )
- Cgive any C and T as above, the ratio  Area  ( C )  Area  ( T ) is independent of C and T
- Dthere exist real numbers a and b such that for any circle C and triangle T as above, we must have a < 
Correct answer
A. given any positive real number α , we can choose C and T as above such that ratio  Area  ( C )
Step-by-step solution
It is given that circumference of C is l and the perimeter of triangle T is l   . Now, let the radius of circle C is r, so 2 π r = l ⇒ r = l 2 π ∴ area of circle C is A 1 = π r 2 = l 2 4 π Now, as we know that area of triangle will be maximum for given perimeter if it is an equilateral triangle, let the length of side of equilateral triangle is ' a, then 3 a = l ⇒ a = l 3 and area of equilateral triangle is A 2 = 3 4 a 2 So,  A 2 = 3 4 l 2 9 = l 2 12 3 ∵ A 1