5 6 9 triangle

Obtuse scalene triangle.

Sides: a = 5   b = 6   c = 9

Area: T = 14.14221356237
Perimeter: p = 20
Semiperimeter: s = 10

Angle ∠ A = α = 31.58663380965° = 31°35'11″ = 0.55112855984 rad
Angle ∠ B = β = 38.9422441269° = 38°56'33″ = 0.68796738189 rad
Angle ∠ C = γ = 109.4711220634° = 109°28'16″ = 1.91106332362 rad

Height: ha = 5.65768542495
Height: hb = 4.71440452079
Height: hc = 3.14326968053

Median: ma = 7.22884161474
Median: mb = 6.63332495807
Median: mc = 3.20215621187

Inradius: r = 1.41442135624
Circumradius: R = 4.7732970773

Vertex coordinates: A[9; 0] B[0; 0] C[3.88988888889; 3.14326968053]
Centroid: CG[4.29662962963; 1.04875656018]
Coordinates of the circumscribed circle: U[4.5; -1.59109902577]
Coordinates of the inscribed circle: I[4; 1.41442135624]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 148.4143661903° = 148°24'49″ = 0.55112855984 rad
∠ B' = β' = 141.0587558731° = 141°3'27″ = 0.68796738189 rad
∠ C' = γ' = 70.52987793655° = 70°31'44″ = 1.91106332362 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 = 5 ; ; b = 6 ; ; c = 9 ; ;

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

p = a+b+c = 5+6+9 = 20 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 20 }{ 2 } = 10 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 10 * (10-5)(10-6)(10-9) } ; ; T = sqrt{ 200 } = 14.14 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 14.14 }{ 5 } = 5.66 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 14.14 }{ 6 } = 4.71 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 14.14 }{ 9 } = 3.14 ; ;

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{ 5**2-6**2-9**2 }{ 2 * 6 * 9 } ) = 31° 35'11" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 6**2-5**2-9**2 }{ 2 * 5 * 9 } ) = 38° 56'33" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 9**2-5**2-6**2 }{ 2 * 6 * 5 } ) = 109° 28'16" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 14.14 }{ 10 } = 1.41 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 5 }{ 2 * sin 31° 35'11" } = 4.77 ; ;




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