9 23 24 triangle

Acute scalene triangle.

Sides: a = 9   b = 23   c = 24

Area: T = 103.1550375666
Perimeter: p = 56
Semiperimeter: s = 28

Angle ∠ A = α = 21.94660412755° = 21°56'46″ = 0.3833030678 rad
Angle ∠ B = β = 72.76547147429° = 72°45'53″ = 1.27699838515 rad
Angle ∠ C = γ = 85.28992439816° = 85°17'21″ = 1.4898578124 rad

Height: ha = 22.92223057035
Height: hb = 8.9769597884
Height: hc = 8.59658646388

Median: ma = 23.07105439901
Median: mb = 14.00989257261
Median: mc = 12.68985775404

Inradius: r = 3.68439419881
Circumradius: R = 12.04106735505

Vertex coordinates: A[24; 0] B[0; 0] C[2.66766666667; 8.59658646388]
Centroid: CG[8.88988888889; 2.86552882129]
Coordinates of the circumscribed circle: U[12; 0.98988475863]
Coordinates of the inscribed circle: I[5; 3.68439419881]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 158.0543958724° = 158°3'14″ = 0.3833030678 rad
∠ B' = β' = 107.2355285257° = 107°14'7″ = 1.27699838515 rad
∠ C' = γ' = 94.71107560184° = 94°42'39″ = 1.4898578124 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 = 23 ; ; c = 24 ; ;

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

p = a+b+c = 9+23+24 = 56 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 56 }{ 2 } = 28 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 28 * (28-9)(28-23)(28-24) } ; ; T = sqrt{ 10640 } = 103.15 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 103.15 }{ 9 } = 22.92 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 103.15 }{ 23 } = 8.97 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 103.15 }{ 24 } = 8.6 ; ;

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{ 9**2-23**2-24**2 }{ 2 * 23 * 24 } ) = 21° 56'46" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 23**2-9**2-24**2 }{ 2 * 9 * 24 } ) = 72° 45'53" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 24**2-9**2-23**2 }{ 2 * 23 * 9 } ) = 85° 17'21" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 103.15 }{ 28 } = 3.68 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9 }{ 2 * sin 21° 56'46" } = 12.04 ; ;




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