9 21 24 triangle

Obtuse scalene triangle.

Sides: a = 9   b = 21   c = 24

Area: T = 93.53107436087
Perimeter: p = 54
Semiperimeter: s = 27

Angle ∠ A = α = 21.78767892983° = 21°47'12″ = 0.38802512067 rad
Angle ∠ B = β = 60° = 1.04771975512 rad
Angle ∠ C = γ = 98.21332107017° = 98°12'48″ = 1.71441438957 rad

Height: ha = 20.78546096908
Height: hb = 8.90876898675
Height: hc = 7.79442286341

Median: ma = 22.0966379794
Median: mb = 14.77332867027
Median: mc = 10.81766538264

Inradius: r = 3.46441016151
Circumradius: R = 12.1244355653

Vertex coordinates: A[24; 0] B[0; 0] C[4.5; 7.79442286341]
Centroid: CG[9.5; 2.59880762114]
Coordinates of the circumscribed circle: U[12; -1.73220508076]
Coordinates of the inscribed circle: I[6; 3.46441016151]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 158.2133210702° = 158°12'48″ = 0.38802512067 rad
∠ B' = β' = 120° = 1.04771975512 rad
∠ C' = γ' = 81.78767892983° = 81°47'12″ = 1.71441438957 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 = 21 ; ; c = 24 ; ;

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

p = a+b+c = 9+21+24 = 54 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 54 }{ 2 } = 27 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 27 * (27-9)(27-21)(27-24) } ; ; T = sqrt{ 8748 } = 93.53 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 93.53 }{ 9 } = 20.78 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 93.53 }{ 21 } = 8.91 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 93.53 }{ 24 } = 7.79 ; ;

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**2-21**2-24**2 }{ 2 * 21 * 24 } ) = 21° 47'12" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-9**2-24**2 }{ 2 * 9 * 24 } ) = 60° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 24**2-9**2-21**2 }{ 2 * 21 * 9 } ) = 98° 12'48" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 93.53 }{ 27 } = 3.46 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9 }{ 2 * sin 21° 47'12" } = 12.12 ; ;




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