8 9 12 triangle

Acute scalene triangle.

Sides: a = 8   b = 9   c = 12

Area: T = 35.9999131934
Perimeter: p = 29
Semiperimeter: s = 14.5

Angle ∠ A = α = 41.80990791939° = 41°48'33″ = 0.73297060892 rad
Angle ∠ B = β = 48.58988113619° = 48°35'20″ = 0.84880347379 rad
Angle ∠ C = γ = 89.60221094442° = 89°36'8″ = 1.56438518265 rad

Height: ha = 98.9997829835
Height: hb = 87.9998070964
Height: hc = 65.9998553223

Median: ma = 9.82334413522
Median: mb = 9.15215026089
Median: mc = 6.04215229868

Inradius: r = 2.48326987541
Circumradius: R = 66.0001446812

Vertex coordinates: A[12; 0] B[0; 0] C[5.29216666667; 65.9998553223]
Centroid: CG[5.76438888889; 21.9999517741]
Coordinates of the circumscribed circle: U[6; 0.04216676714]
Coordinates of the inscribed circle: I[5.5; 2.48326987541]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 138.1910920806° = 138°11'27″ = 0.73297060892 rad
∠ B' = β' = 131.4111188638° = 131°24'40″ = 0.84880347379 rad
∠ C' = γ' = 90.39878905558° = 90°23'52″ = 1.56438518265 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 = 8 ; ; b = 9 ; ; c = 12 ; ;

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

p = a+b+c = 8+9+12 = 29 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 29 }{ 2 } = 14.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 14.5 * (14.5-8)(14.5-9)(14.5-12) } ; ; T = sqrt{ 1295.94 } = 36 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 36 }{ 8 } = 9 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 36 }{ 9 } = 8 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 36 }{ 12 } = 6 ; ;

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{ 8**2-9**2-12**2 }{ 2 * 9 * 12 } ) = 41° 48'33" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 9**2-8**2-12**2 }{ 2 * 8 * 12 } ) = 48° 35'20" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 12**2-8**2-9**2 }{ 2 * 9 * 8 } ) = 89° 36'8" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 36 }{ 14.5 } = 2.48 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8 }{ 2 * sin 41° 48'33" } = 6 ; ;




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