5 19 21 triangle

Obtuse scalene triangle.

Sides: a = 5   b = 19   c = 21

Area: T = 45.46663336987
Perimeter: p = 45
Semiperimeter: s = 22.5

Angle ∠ A = α = 13.17435511073° = 13°10'25″ = 0.2329921841 rad
Angle ∠ B = β = 60° = 1.04771975512 rad
Angle ∠ C = γ = 106.8266448893° = 106°49'35″ = 1.86444732614 rad

Height: ha = 18.18765334795
Height: hb = 4.7865929863
Height: hc = 4.33301270189

Median: ma = 19.86883164863
Median: mb = 11.94878031453
Median: mc = 9.09767026993

Inradius: r = 2.02107259422
Circumradius: R = 10.97696551146

Vertex coordinates: A[21; 0] B[0; 0] C[2.5; 4.33301270189]
Centroid: CG[7.83333333333; 1.4433375673]
Coordinates of the circumscribed circle: U[10.5; -3.17554264805]
Coordinates of the inscribed circle: I[3.5; 2.02107259422]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 166.8266448893° = 166°49'35″ = 0.2329921841 rad
∠ B' = β' = 120° = 1.04771975512 rad
∠ C' = γ' = 73.17435511073° = 73°10'25″ = 1.86444732614 rad

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How did we calculate this triangle?

Now we know the lengths of all three sides of the triangle and the triangle is uniquely determined. Next we calculate another its characteristics - same procedure as calculation of the triangle from the known three sides SSS.

a = 5 ; ; b = 19 ; ; c = 21 ; ;

1. The triangle circumference is the sum of the lengths of its three sides

p = a+b+c = 5+19+21 = 45 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 45 }{ 2 } = 22.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 22.5 * (22.5-5)(22.5-19)(22.5-21) } ; ; T = sqrt{ 2067.19 } = 45.47 ; ;

4. Calculate the heights of the triangle from its area.

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 45.47 }{ 5 } = 18.19 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 45.47 }{ 19 } = 4.79 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 45.47 }{ 21 } = 4.33 ; ;

5. Calculation of the inner angles of the triangle using a Law of Cosines

a**2 = b**2+c**2 - 2bc cos( alpha ) ; ; alpha = arccos( fraction{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 5**2-19**2-21**2 }{ 2 * 19 * 21 } ) = 13° 10'25" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 19**2-5**2-21**2 }{ 2 * 5 * 21 } ) = 60° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-5**2-19**2 }{ 2 * 19 * 5 } ) = 106° 49'35" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 45.47 }{ 22.5 } = 2.02 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 5 }{ 2 * sin 13° 10'25" } = 10.97 ; ;




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