10 17 24 triangle

Obtuse scalene triangle.

Sides: a = 10   b = 17   c = 24

Area: T = 70.98989956261
Perimeter: p = 51
Semiperimeter: s = 25.5

Angle ∠ A = α = 20.36441348063° = 20°21'51″ = 0.35554212017 rad
Angle ∠ B = β = 36.26988522245° = 36°16'8″ = 0.63330108872 rad
Angle ∠ C = γ = 123.3677012969° = 123°22'1″ = 2.15331605647 rad

Height: ha = 14.19877991252
Height: hb = 8.35216465442
Height: hc = 5.91657496355

Median: ma = 20.18766292382
Median: mb = 16.30218403869
Median: mc = 7.10663352018

Inradius: r = 2.78438821814
Circumradius: R = 14.36884241621

Vertex coordinates: A[24; 0] B[0; 0] C[8.06325; 5.91657496355]
Centroid: CG[10.68875; 1.97219165452]
Coordinates of the circumscribed circle: U[12; -7.90326332892]
Coordinates of the inscribed circle: I[8.5; 2.78438821814]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 159.6365865194° = 159°38'9″ = 0.35554212017 rad
∠ B' = β' = 143.7311147776° = 143°43'52″ = 0.63330108872 rad
∠ C' = γ' = 56.63329870308° = 56°37'59″ = 2.15331605647 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 = 10 ; ; b = 17 ; ; c = 24 ; ;

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

p = a+b+c = 10+17+24 = 51 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 51 }{ 2 } = 25.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 25.5 * (25.5-10)(25.5-17)(25.5-24) } ; ; T = sqrt{ 5039.44 } = 70.99 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 70.99 }{ 10 } = 14.2 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 70.99 }{ 17 } = 8.35 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 70.99 }{ 24 } = 5.92 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 17**2+24**2-10**2 }{ 2 * 17 * 24 } ) = 20° 21'51" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 10**2+24**2-17**2 }{ 2 * 10 * 24 } ) = 36° 16'8" ; ; gamma = 180° - alpha - beta = 180° - 20° 21'51" - 36° 16'8" = 123° 22'1" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 70.99 }{ 25.5 } = 2.78 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 10 }{ 2 * sin 20° 21'51" } = 14.37 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 17**2+2 * 24**2 - 10**2 } }{ 2 } = 20.187 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 24**2+2 * 10**2 - 17**2 } }{ 2 } = 16.302 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 17**2+2 * 10**2 - 24**2 } }{ 2 } = 7.106 ; ;
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