21 24 24 triangle

Acute isosceles triangle.

Sides: a = 21   b = 24   c = 24

Area: T = 226.6033039476
Perimeter: p = 69
Semiperimeter: s = 34.5

Angle ∠ A = α = 51.88989595447° = 51°53'20″ = 0.90656331895 rad
Angle ∠ B = β = 64.05655202276° = 64°3'20″ = 1.1187979732 rad
Angle ∠ C = γ = 64.05655202276° = 64°3'20″ = 1.1187979732 rad

Height: ha = 21.58112418549
Height: hb = 18.8843586623
Height: hc = 18.8843586623

Median: ma = 21.58112418549
Median: mb = 19.0921883092
Median: mc = 19.0921883092

Inradius: r = 6.56882040428
Circumradius: R = 13.34549224997

Vertex coordinates: A[24; 0] B[0; 0] C[9.18875; 18.8843586623]
Centroid: CG[11.06325; 6.29545288743]
Coordinates of the circumscribed circle: U[12; 5.83884035936]
Coordinates of the inscribed circle: I[10.5; 6.56882040428]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 128.1111040455° = 128°6'40″ = 0.90656331895 rad
∠ B' = β' = 115.9444479772° = 115°56'40″ = 1.1187979732 rad
∠ C' = γ' = 115.9444479772° = 115°56'40″ = 1.1187979732 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 = 21 ; ; b = 24 ; ; c = 24 ; ;

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

p = a+b+c = 21+24+24 = 69 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 69 }{ 2 } = 34.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 34.5 * (34.5-21)(34.5-24)(34.5-24) } ; ; T = sqrt{ 51348.94 } = 226.6 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 226.6 }{ 21 } = 21.58 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 226.6 }{ 24 } = 18.88 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 226.6 }{ 24 } = 18.88 ; ;

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{ 21**2-24**2-24**2 }{ 2 * 24 * 24 } ) = 51° 53'20" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 24**2-21**2-24**2 }{ 2 * 21 * 24 } ) = 64° 3'20" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 24**2-21**2-24**2 }{ 2 * 24 * 21 } ) = 64° 3'20" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 226.6 }{ 34.5 } = 6.57 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 21 }{ 2 * sin 51° 53'20" } = 13.34 ; ;




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