9 24 25 triangle

Acute scalene triangle.

Sides: a = 9   b = 24   c = 25

Area: T = 107.7033296143
Perimeter: p = 58
Semiperimeter: s = 29

Angle ∠ A = α = 21.03994697813° = 21°2'22″ = 0.36772080206 rad
Angle ∠ B = β = 73.20985528854° = 73°12'31″ = 1.27877302885 rad
Angle ∠ C = γ = 85.75219773333° = 85°45'7″ = 1.49766543446 rad

Height: ha = 23.93440658095
Height: hb = 8.97552746786
Height: hc = 8.61662636914

Median: ma = 24.08883789409
Median: mb = 14.45768322948
Median: mc = 13.12444047484

Inradius: r = 3.71439067635
Circumradius: R = 12.5344435327

Vertex coordinates: A[25; 0] B[0; 0] C[2.6; 8.61662636914]
Centroid: CG[9.2; 2.87220878971]
Coordinates of the circumscribed circle: U[12.5; 0.92884766909]
Coordinates of the inscribed circle: I[5; 3.71439067635]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 158.9610530219° = 158°57'38″ = 0.36772080206 rad
∠ B' = β' = 106.7911447115° = 106°47'29″ = 1.27877302885 rad
∠ C' = γ' = 94.24880226667° = 94°14'53″ = 1.49766543446 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 = 24 ; ; c = 25 ; ;

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

p = a+b+c = 9+24+25 = 58 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 58 }{ 2 } = 29 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 29 * (29-9)(29-24)(29-25) } ; ; T = sqrt{ 11600 } = 107.7 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 107.7 }{ 9 } = 23.93 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 107.7 }{ 24 } = 8.98 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 107.7 }{ 25 } = 8.62 ; ;

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**2-24**2-25**2 }{ 2 * 24 * 25 } ) = 21° 2'22" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 24**2-9**2-25**2 }{ 2 * 9 * 25 } ) = 73° 12'31" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 25**2-9**2-24**2 }{ 2 * 24 * 9 } ) = 85° 45'7" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 107.7 }{ 29 } = 3.71 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9 }{ 2 * sin 21° 2'22" } = 12.53 ; ;




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