17 19 24 triangle

Acute scalene triangle.

Sides: a = 17   b = 19   c = 24

Area: T = 160.4376903486
Perimeter: p = 60
Semiperimeter: s = 30

Angle ∠ A = α = 44.72222460248° = 44°43'20″ = 0.7810550442 rad
Angle ∠ B = β = 51.85554868988° = 51°51'20″ = 0.90550489816 rad
Angle ∠ C = γ = 83.42222670764° = 83°25'20″ = 1.456599323 rad

Height: ha = 18.87549298218
Height: hb = 16.88880951037
Height: hc = 13.37697419571

Median: ma = 19.90660292374
Median: mb = 18.5
Median: mc = 13.45436240471

Inradius: r = 5.34878967828
Circumradius: R = 12.08795151109

Vertex coordinates: A[24; 0] B[0; 0] C[10.5; 13.37697419571]
Centroid: CG[11.5; 4.45765806524]
Coordinates of the circumscribed circle: U[12; 1.38437215452]
Coordinates of the inscribed circle: I[11; 5.34878967828]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135.2787753975° = 135°16'40″ = 0.7810550442 rad
∠ B' = β' = 128.1454513101° = 128°8'40″ = 0.90550489816 rad
∠ C' = γ' = 96.57877329236° = 96°34'40″ = 1.456599323 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 = 17 ; ; b = 19 ; ; c = 24 ; ;

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

p = a+b+c = 17+19+24 = 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 * (30-17)(30-19)(30-24) } ; ; T = sqrt{ 25740 } = 160.44 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 160.44 }{ 17 } = 18.87 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 160.44 }{ 19 } = 16.89 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 160.44 }{ 24 } = 13.37 ; ;

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{ 17**2-19**2-24**2 }{ 2 * 19 * 24 } ) = 44° 43'20" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 19**2-17**2-24**2 }{ 2 * 17 * 24 } ) = 51° 51'20" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 24**2-17**2-19**2 }{ 2 * 19 * 17 } ) = 83° 25'20" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 160.44 }{ 30 } = 5.35 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 17 }{ 2 * sin 44° 43'20" } = 12.08 ; ;




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