10 13 19 triangle

Obtuse scalene triangle.

Sides: a = 10   b = 13   c = 19

Area: T = 60.79547366143
Perimeter: p = 42
Semiperimeter: s = 21

Angle ∠ A = α = 29.49895673359° = 29°29'22″ = 0.5154690045 rad
Angle ∠ B = β = 39.78876882825° = 39°47'16″ = 0.69444261623 rad
Angle ∠ C = γ = 110.7232744382° = 110°43'22″ = 1.93224764463 rad

Height: ha = 12.15989473229
Height: hb = 9.35330364022
Height: hc = 6.39994459594

Median: ma = 15.49219333848
Median: mb = 13.7220422734
Median: mc = 6.65220673478

Inradius: r = 2.89549874578
Circumradius: R = 10.15771292909

Vertex coordinates: A[19; 0] B[0; 0] C[7.68442105263; 6.39994459594]
Centroid: CG[8.89547368421; 2.13331486531]
Coordinates of the circumscribed circle: U[9.5; -3.59440611337]
Coordinates of the inscribed circle: I[8; 2.89549874578]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 150.5110432664° = 150°30'38″ = 0.5154690045 rad
∠ B' = β' = 140.2122311717° = 140°12'44″ = 0.69444261623 rad
∠ C' = γ' = 69.27772556184° = 69°16'38″ = 1.93224764463 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 = 10 ; ; b = 13 ; ; c = 19 ; ;

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

p = a+b+c = 10+13+19 = 42 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 42 }{ 2 } = 21 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 21 * (21-10)(21-13)(21-19) } ; ; T = sqrt{ 3696 } = 60.79 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 60.79 }{ 10 } = 12.16 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 60.79 }{ 13 } = 9.35 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 60.79 }{ 19 } = 6.4 ; ;

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{ 10**2-13**2-19**2 }{ 2 * 13 * 19 } ) = 29° 29'22" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 13**2-10**2-19**2 }{ 2 * 10 * 19 } ) = 39° 47'16" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 19**2-10**2-13**2 }{ 2 * 13 * 10 } ) = 110° 43'22" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 60.79 }{ 21 } = 2.89 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 10 }{ 2 * sin 29° 29'22" } = 10.16 ; ;




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