Question

The perimeter of a triangle is 584 cm. The interior angles measure 50°, 60° and 70°, respectively. What is the radius of the largest circle that can be cut from this triangle in centimeters?

Answer 1

Answer:

triangles with these angles are similar, and P is directly related to the picked side length.

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L = 10

Perimeter = 33.57197472

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Multiply L by 168/P

a 55.40335354

b 62.6343707

c 67.96227419

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Perimeter = 185.9999984

Area = 1630.43914

Answer 2

The radius of the largest circle that can be cut from the given triangle is 55 cm.

The perimeter of a triangle is 584 cm. The interior angles measure 50°, 60° and 70°, respectively.

We have to find the radius of the largest circle that can be cut from this triangle in centimetres.

Let a , b and c are sides of triangle.

now using Sine rule

$\frac{sinA}{a}=\frac{sinB}{b}=\frac{sinC}{c}$

here, A = 50° , B = 60° and C = 70°

∴ sin50°/a = sin60°/b = sin70°/c = k (let)

⇒a = sin50°/k , b = sin60°/k , c = sin70°/k

perimeter of triangle, a + b + c = 584

⇒sin50°/k + sin60°/k + sin70°/k = 584

⇒k = (sin50° + sin60° + sin70°)/584

Incircle is the largest circle that can be cut from this triangle. and it is found by the ratio of area of triangle to semiperimeter.

i.e., r = ∆/s

we know, area of triangle = 1/2 ab sinC = 1/2 bc sinA = 1/2 ca sinB

we can choose any of one

∴ ∆ = 1/2 ab sinC = 1/2 (sin50°/k)(sin60°/k) sin70°

= (sin50° sin60° sin70°)/2k²

= {584²(sin50° sin60° sin70°)}/{2(sin50° + sin60° + sin70°)²}

now Radius , r = {584²(sin50° sin60° sin70°)}/{2(sin50° + sin60° + sin70°)² × 584/2}

= {584(sin50° sin60° sin70°)}/(sin50° + sin60° + sin70°)²

= 55.04 cm ≈ 55 cm

Therefore the radius of the largest circle that can be cut from the given triangle is 55 cm.