10 28 29 triangle

Acute scalene triangle.

Sides: a = 10   b = 28   c = 29

Area: T = 139.5876666627
Perimeter: p = 67
Semiperimeter: s = 33.5

Angle ∠ A = α = 20.10991416034° = 20°6'33″ = 0.35109707307 rad
Angle ∠ B = β = 74.29546905361° = 74°17'41″ = 1.29766869666 rad
Angle ∠ C = γ = 85.59661678604° = 85°35'46″ = 1.49439349563 rad

Height: ha = 27.91773333254
Height: hb = 9.97704761876
Height: hc = 9.62766666639

Median: ma = 28.06224304008
Median: mb = 16.56880415258
Median: mc = 15.22333373476

Inradius: r = 4.1676766168
Circumradius: R = 14.54329362922

Vertex coordinates: A[29; 0] B[0; 0] C[2.70768965517; 9.62766666639]
Centroid: CG[10.56989655172; 3.2098888888]
Coordinates of the circumscribed circle: U[14.5; 1.1176689751]
Coordinates of the inscribed circle: I[5.5; 4.1676766168]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 159.8910858397° = 159°53'27″ = 0.35109707307 rad
∠ B' = β' = 105.7055309464° = 105°42'19″ = 1.29766869666 rad
∠ C' = γ' = 94.40438321396° = 94°24'14″ = 1.49439349563 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 = 28 ; ; c = 29 ; ;

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

p = a+b+c = 10+28+29 = 67 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 67 }{ 2 } = 33.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 33.5 * (33.5-10)(33.5-28)(33.5-29) } ; ; T = sqrt{ 19484.44 } = 139.59 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 139.59 }{ 10 } = 27.92 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 139.59 }{ 28 } = 9.97 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 139.59 }{ 29 } = 9.63 ; ;

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{ 28**2+29**2-10**2 }{ 2 * 28 * 29 } ) = 20° 6'33" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 10**2+29**2-28**2 }{ 2 * 10 * 29 } ) = 74° 17'41" ; ; gamma = 180° - alpha - beta = 180° - 20° 6'33" - 74° 17'41" = 85° 35'46" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 139.59 }{ 33.5 } = 4.17 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 10 }{ 2 * sin 20° 6'33" } = 14.54 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 28**2+2 * 29**2 - 10**2 } }{ 2 } = 28.062 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 29**2+2 * 10**2 - 28**2 } }{ 2 } = 16.568 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 28**2+2 * 10**2 - 29**2 } }{ 2 } = 15.223 ; ;
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