19 19 22 triangle

Acute isosceles triangle.

Sides: a = 19   b = 19   c = 22

Area: T = 170.4111267233
Perimeter: p = 60
Semiperimeter: s = 30

Angle ∠ A = α = 54.62334598481° = 54°37'24″ = 0.95333592232 rad
Angle ∠ B = β = 54.62334598481° = 54°37'24″ = 0.95333592232 rad
Angle ∠ C = γ = 70.75330803039° = 70°45'11″ = 1.23548742072 rad

Height: ha = 17.93880281298
Height: hb = 17.93880281298
Height: hc = 15.49219333848

Median: ma = 18.22877261336
Median: mb = 18.22877261336
Median: mc = 15.49219333848

Inradius: r = 5.68803755744
Circumradius: R = 11.65112248998

Vertex coordinates: A[22; 0] B[0; 0] C[11; 15.49219333848]
Centroid: CG[11; 5.16439777949]
Coordinates of the circumscribed circle: U[11; 3.8410708485]
Coordinates of the inscribed circle: I[11; 5.68803755744]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 125.3776540152° = 125°22'36″ = 0.95333592232 rad
∠ B' = β' = 125.3776540152° = 125°22'36″ = 0.95333592232 rad
∠ C' = γ' = 109.2476919696° = 109°14'49″ = 1.23548742072 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 = 19 ; ; b = 19 ; ; c = 22 ; ;

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

p = a+b+c = 19+19+22 = 60 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 60 }{ 2 } = 30 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 30 * (30-19)(30-19)(30-22) } ; ; T = sqrt{ 29040 } = 170.41 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 170.41 }{ 19 } = 17.94 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 170.41 }{ 19 } = 17.94 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 170.41 }{ 22 } = 15.49 ; ;

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{ 19**2-19**2-22**2 }{ 2 * 19 * 22 } ) = 54° 37'24" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 19**2-19**2-22**2 }{ 2 * 19 * 22 } ) = 54° 37'24" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 22**2-19**2-19**2 }{ 2 * 19 * 19 } ) = 70° 45'11" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 170.41 }{ 30 } = 5.68 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 19 }{ 2 * sin 54° 37'24" } = 11.65 ; ;




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