Question

The ratio of areas of incircle and circumcircle of an isosceles right angled triangle is

Answer 1

we know, formula of incircle radius , r = ∆/s
where, ∆ = area of triangle
s = semiperimeter of triangle

For equilateral triangle ,
all sides are equal.
Let side length = a
Then, area of equilateral triangle , ∆ = √3/4 a²
semi-perimeter of equilateral triangle , S = 3a/2
Now, r = √3/4 × a/{3a/2} = a/2√3

circumcircle radius is given by
2R = a/sinA = b/sinB = c/sinC
Here, R is the circumcircle radius
a , b, c are sides of triangle and A , B, C are angle of triangle.
For equilateral triangle,
a = b = c and A = B = C = 60°
so, 2R = a/sin60° = b/sin60° = c/sin60°
Take anyone of it
2R = a/sin60°
⇒R = a/2sin60° = a/√3 [ ∵ sin60° = √3/2 ]

Now,ratio of area incircle and circumcircle = πr²/πR²
= r²/R²
= (a/2√3)²/(a/√3)²
= a² × 3/a² × 4 × 3
= 1/4

Hence, answer is 1 : 4