Question

If delta1 and delta2 are the areas of incircle and cirucumcircle of a triangle with sides 3,4 and 5 then delta1/delta2=?
Answer with full steps

Answer 1

In radius =area of triangle /semi perimeter
r=1/2*3*4/6
r=1
circum radius =half of hypotenuse
R=2.5
delta 1/delta 2=πr^2/πR^2=1/6.25=4/25

Answer 2

The Value of Δ₁/Δ₂ comes out to be 4/25.

Given,

Δ₁ =  Area of the incircle

Δ₂ = Area of the circumcircle

Sides of triangle → 3, 4 and 5

To Find,

Value of Δ₁/Δ₂

Solution,

If we observe carefully, it can be noticed that the given sides are that of a right-angled triangle.

We can prove that by the Pythagoras theorem

P² + B² = H²

3² + 4² = 5²

9 + 16 = 25

25 = 25

Now, we know that the formula for the area of incircle for a right-angled triangle is given by -

Area of Incircle = π/4 x (P + B - H)²

where P, B & H stands for perpendicular, base, and hypotenuse of the triangle respectively

Δ₁ = π/4 x (3 + 4 - 5)²

Δ₁ = π/4 x (2)²

Δ₁ = π/4 x 4

Δ₁ = π

Also, we know that the formula for the area of circumcircle for a right-angled triangle is given by -

Area of circumcircle = π/4 x H²

where H stands for the hypotenuse of the triangle

Δ₂ = π/4 x 5²

Δ₂ = π/4 x 25

Δ₂ = 25π/4

We finally need the value of Δ₁/Δ₂

Δ₁/Δ₂ = (π) / (25π/4)

Δ₁/Δ₂ = 4/25

Thus, the ratio comes out to be 4/25.

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