9 17 20 triangle

Obtuse scalene triangle.

Sides: a = 9   b = 17   c = 20

Area: T = 76.13114652427
Perimeter: p = 46
Semiperimeter: s = 23

Angle ∠ A = α = 26.60546816894° = 26°36'17″ = 0.46443392919 rad
Angle ∠ B = β = 57.76990473645° = 57°46'9″ = 1.00882600823 rad
Angle ∠ C = γ = 95.62662709461° = 95°37'35″ = 1.66989932794 rad

Height: ha = 16.91881033873
Height: hb = 8.95766429697
Height: hc = 7.61331465243

Median: ma = 18.00769431054
Median: mb = 12.97111217711
Median: mc = 9.22195444573

Inradius: r = 3.31100637062
Circumradius: R = 10.04884076795

Vertex coordinates: A[20; 0] B[0; 0] C[4.8; 7.61331465243]
Centroid: CG[8.26766666667; 2.53877155081]
Coordinates of the circumscribed circle: U[10; -0.98551380078]
Coordinates of the inscribed circle: I[6; 3.31100637062]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 153.3955318311° = 153°23'43″ = 0.46443392919 rad
∠ B' = β' = 122.2310952636° = 122°13'51″ = 1.00882600823 rad
∠ C' = γ' = 84.37437290539° = 84°22'25″ = 1.66989932794 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 = 9 ; ; b = 17 ; ; c = 20 ; ;

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

p = a+b+c = 9+17+20 = 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 * (23-9)(23-17)(23-20) } ; ; T = sqrt{ 5796 } = 76.13 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 76.13 }{ 9 } = 16.92 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 76.13 }{ 17 } = 8.96 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 76.13 }{ 20 } = 7.61 ; ;

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{ 9**2-17**2-20**2 }{ 2 * 17 * 20 } ) = 26° 36'17" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 17**2-9**2-20**2 }{ 2 * 9 * 20 } ) = 57° 46'9" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 20**2-9**2-17**2 }{ 2 * 17 * 9 } ) = 95° 37'35" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 76.13 }{ 23 } = 3.31 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9 }{ 2 * sin 26° 36'17" } = 10.05 ; ;




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