40 75 85 triangle

Right scalene triangle.

Sides: a = 40   b = 75   c = 85

Area: T = 1500
Perimeter: p = 200
Semiperimeter: s = 100

Angle ∠ A = α = 28.07224869359° = 28°4'21″ = 0.49899573263 rad
Angle ∠ B = β = 61.92875130641° = 61°55'39″ = 1.08108390005 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 75
Height: hb = 40
Height: hc = 35.29441176471

Median: ma = 77.62108734813
Median: mb = 54.82992804987
Median: mc = 42.5

Inradius: r = 15
Circumradius: R = 42.5

Vertex coordinates: A[85; 0] B[0; 0] C[18.82435294118; 35.29441176471]
Centroid: CG[34.60878431373; 11.76547058824]
Coordinates of the circumscribed circle: U[42.5; 0]
Coordinates of the inscribed circle: I[25; 15]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 151.9287513064° = 151°55'39″ = 0.49899573263 rad
∠ B' = β' = 118.0722486936° = 118°4'21″ = 1.08108390005 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

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

p = a+b+c = 40+75+85 = 200 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 200 }{ 2 } = 100 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 100 * (100-40)(100-75)(100-85) } ; ; T = sqrt{ 2250000 } = 1500 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1500 }{ 40 } = 75 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1500 }{ 75 } = 40 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1500 }{ 85 } = 35.29 ; ;

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{ 75**2+85**2-40**2 }{ 2 * 75 * 85 } ) = 28° 4'21" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 40**2+85**2-75**2 }{ 2 * 40 * 85 } ) = 61° 55'39" ; ;
 gamma = 180° - alpha - beta = 180° - 28° 4'21" - 61° 55'39" = 90° ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1500 }{ 100 } = 15 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 40 }{ 2 * sin 28° 4'21" } = 42.5 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 75**2+2 * 85**2 - 40**2 } }{ 2 } = 77.621 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 85**2+2 * 40**2 - 75**2 } }{ 2 } = 54.829 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 75**2+2 * 40**2 - 85**2 } }{ 2 } = 42.5 ; ;
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