9 24 24 triangle

Acute isosceles triangle.

Sides: a = 9   b = 24   c = 24

Area: T = 106.0854577107
Perimeter: p = 57
Semiperimeter: s = 28.5

Angle ∠ A = α = 21.61438457497° = 21°36'50″ = 0.37772327724 rad
Angle ∠ B = β = 79.19330771251° = 79°11'35″ = 1.38221799406 rad
Angle ∠ C = γ = 79.19330771251° = 79°11'35″ = 1.38221799406 rad

Height: ha = 23.57443504683
Height: hb = 8.84403814256
Height: hc = 8.84403814256

Median: ma = 23.57443504683
Median: mb = 13.58330777072
Median: mc = 13.58330777072

Inradius: r = 3.72222658634
Circumradius: R = 12.21766674491

Vertex coordinates: A[24; 0] B[0; 0] C[1.68875; 8.84403814256]
Centroid: CG[8.56325; 2.94767938085]
Coordinates of the circumscribed circle: U[12; 2.29106251467]
Coordinates of the inscribed circle: I[4.5; 3.72222658634]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 158.386615425° = 158°23'10″ = 0.37772327724 rad
∠ B' = β' = 100.8076922875° = 100°48'25″ = 1.38221799406 rad
∠ C' = γ' = 100.8076922875° = 100°48'25″ = 1.38221799406 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 = 24 ; ;

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

p = a+b+c = 9+24+24 = 57 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 57 }{ 2 } = 28.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 28.5 * (28.5-9)(28.5-24)(28.5-24) } ; ; T = sqrt{ 11253.94 } = 106.08 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 106.08 }{ 9 } = 23.57 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 106.08 }{ 24 } = 8.84 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 106.08 }{ 24 } = 8.84 ; ;

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**2-24**2 }{ 2 * 24 * 24 } ) = 21° 36'50" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 24**2-9**2-24**2 }{ 2 * 9 * 24 } ) = 79° 11'35" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 24**2-9**2-24**2 }{ 2 * 24 * 9 } ) = 79° 11'35" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 106.08 }{ 28.5 } = 3.72 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9 }{ 2 * sin 21° 36'50" } = 12.22 ; ;




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