10 12 19 triangle

Obtuse scalene triangle.

Sides: a = 10   b = 12   c = 19

Area: T = 52.38773792053
Perimeter: p = 41
Semiperimeter: s = 20.5

Angle ∠ A = α = 27.35773262181° = 27°21'26″ = 0.4777475417 rad
Angle ∠ B = β = 33.46662779269° = 33°27'59″ = 0.58440967382 rad
Angle ∠ C = γ = 119.1766395855° = 119°10'35″ = 2.08800204983 rad

Height: ha = 10.47774758411
Height: hb = 8.73112298676
Height: hc = 5.5144460969

Median: ma = 15.0833103129
Median: mb = 13.9466325681
Median: mc = 5.63547138348

Inradius: r = 2.55554819125
Circumradius: R = 10.88804832127

Vertex coordinates: A[19; 0] B[0; 0] C[8.34221052632; 5.5144460969]
Centroid: CG[9.11440350877; 1.83881536563]
Coordinates of the circumscribed circle: U[9.5; -5.30442355662]
Coordinates of the inscribed circle: I[8.5; 2.55554819125]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 152.6432673782° = 152°38'34″ = 0.4777475417 rad
∠ B' = β' = 146.5343722073° = 146°32'1″ = 0.58440967382 rad
∠ C' = γ' = 60.8243604145° = 60°49'25″ = 2.08800204983 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 = 12 ; ; c = 19 ; ;

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

p = a+b+c = 10+12+19 = 41 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 41 }{ 2 } = 20.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 20.5 * (20.5-10)(20.5-12)(20.5-19) } ; ; T = sqrt{ 2744.44 } = 52.39 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 52.39 }{ 10 } = 10.48 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 52.39 }{ 12 } = 8.73 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 52.39 }{ 19 } = 5.51 ; ;

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**2-12**2-19**2 }{ 2 * 12 * 19 } ) = 27° 21'26" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 12**2-10**2-19**2 }{ 2 * 10 * 19 } ) = 33° 27'59" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 19**2-10**2-12**2 }{ 2 * 12 * 10 } ) = 119° 10'35" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 52.39 }{ 20.5 } = 2.56 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 10 }{ 2 * sin 27° 21'26" } = 10.88 ; ;




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