12 15 19 triangle

Acute scalene triangle.

Sides: a = 12   b = 15   c = 19

Area: T = 89.97877750336
Perimeter: p = 46
Semiperimeter: s = 23

Angle ∠ A = α = 39.15551858195° = 39°9'19″ = 0.68333869118 rad
Angle ∠ B = β = 52.11881585419° = 52°7'5″ = 0.91096334666 rad
Angle ∠ C = γ = 88.72766556386° = 88°43'36″ = 1.54985722752 rad

Height: ha = 14.99662958389
Height: hb = 11.99770366711
Height: hc = 9.47113447404

Median: ma = 16.03112195419
Median: mb = 14.00989257261
Median: mc = 9.70882439195

Inradius: r = 3.91220771754
Circumradius: R = 9.50223465481

Vertex coordinates: A[19; 0] B[0; 0] C[7.36884210526; 9.47113447404]
Centroid: CG[8.78994736842; 3.15771149135]
Coordinates of the circumscribed circle: U[9.5; 0.21111632566]
Coordinates of the inscribed circle: I[8; 3.91220771754]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 140.845481418° = 140°50'41″ = 0.68333869118 rad
∠ B' = β' = 127.8821841458° = 127°52'55″ = 0.91096334666 rad
∠ C' = γ' = 91.27333443614° = 91°16'24″ = 1.54985722752 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 = 12 ; ; b = 15 ; ; c = 19 ; ;

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

p = a+b+c = 12+15+19 = 46 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 46 }{ 2 } = 23 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 23 * (23-12)(23-15)(23-19) } ; ; T = sqrt{ 8096 } = 89.98 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 89.98 }{ 12 } = 15 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 89.98 }{ 15 } = 12 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 89.98 }{ 19 } = 9.47 ; ;

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{ 12**2-15**2-19**2 }{ 2 * 15 * 19 } ) = 39° 9'19" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 15**2-12**2-19**2 }{ 2 * 12 * 19 } ) = 52° 7'5" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 19**2-12**2-15**2 }{ 2 * 15 * 12 } ) = 88° 43'36" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 89.98 }{ 23 } = 3.91 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 39° 9'19" } = 9.5 ; ;




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