19 19 24 triangle

Acute isosceles triangle.

Sides: a = 19   b = 19   c = 24

Area: T = 176.7711038352
Perimeter: p = 62
Semiperimeter: s = 31

Angle ∠ A = α = 50.83332892839° = 50°50' = 0.88772082676 rad
Angle ∠ B = β = 50.83332892839° = 50°50' = 0.88772082676 rad
Angle ∠ C = γ = 78.33334214322° = 78°20' = 1.36771761183 rad

Height: ha = 18.60774777213
Height: hb = 18.60774777213
Height: hc = 14.73109198627

Median: ma = 19.44986503388
Median: mb = 19.44986503388
Median: mc = 14.73109198627

Inradius: r = 5.70222915597
Circumradius: R = 12.25331384111

Vertex coordinates: A[24; 0] B[0; 0] C[12; 14.73109198627]
Centroid: CG[12; 4.91103066209]
Coordinates of the circumscribed circle: U[12; 2.47877814516]
Coordinates of the inscribed circle: I[12; 5.70222915597]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 129.1676710716° = 129°10' = 0.88772082676 rad
∠ B' = β' = 129.1676710716° = 129°10' = 0.88772082676 rad
∠ C' = γ' = 101.6676578568° = 101°40' = 1.36771761183 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 = 19 ; ; b = 19 ; ; c = 24 ; ;

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

p = a+b+c = 19+19+24 = 62 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 62 }{ 2 } = 31 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 31 * (31-19)(31-19)(31-24) } ; ; T = sqrt{ 31248 } = 176.77 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 176.77 }{ 19 } = 18.61 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 176.77 }{ 19 } = 18.61 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 176.77 }{ 24 } = 14.73 ; ;

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{ 19**2-19**2-24**2 }{ 2 * 19 * 24 } ) = 50° 50' ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 19**2-19**2-24**2 }{ 2 * 19 * 24 } ) = 50° 50' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 24**2-19**2-19**2 }{ 2 * 19 * 19 } ) = 78° 20' ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 176.77 }{ 31 } = 5.7 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 19 }{ 2 * sin 50° 50' } = 12.25 ; ;




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