45 60 75 triangle

Right scalene Pythagorean triangle.

Sides: a = 45   b = 60   c = 75

Area: T = 1350
Perimeter: p = 180
Semiperimeter: s = 90

Angle ∠ A = α = 36.87698976458° = 36°52'12″ = 0.64435011088 rad
Angle ∠ B = β = 53.13301023542° = 53°7'48″ = 0.9277295218 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 60
Height: hb = 45
Height: hc = 36

Median: ma = 64.08800280899
Median: mb = 54.0833269132
Median: mc = 37.5

Inradius: r = 15
Circumradius: R = 37.5

Vertex coordinates: A[75; 0] B[0; 0] C[27; 36]
Centroid: CG[34; 12]
Coordinates of the circumscribed circle: U[37.5; 0]
Coordinates of the inscribed circle: I[30; 15]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 143.1330102354° = 143°7'48″ = 0.64435011088 rad
∠ B' = β' = 126.8769897646° = 126°52'12″ = 0.9277295218 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 = 45+60+75 = 180 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 180 }{ 2 } = 90 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 90 * (90-45)(90-60)(90-75) } ; ; T = sqrt{ 1822500 } = 1350 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1350 }{ 45 } = 60 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1350 }{ 60 } = 45 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1350 }{ 75 } = 36 ; ;

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{ 60**2+75**2-45**2 }{ 2 * 60 * 75 } ) = 36° 52'12" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 45**2+75**2-60**2 }{ 2 * 45 * 75 } ) = 53° 7'48" ; ;
gamma = 180° - alpha - beta = 180° - 36° 52'12" - 53° 7'48" = 90° ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1350 }{ 90 } = 15 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 45 }{ 2 * sin 36° 52'12" } = 37.5 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 60**2+2 * 75**2 - 45**2 } }{ 2 } = 64.08 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 75**2+2 * 45**2 - 60**2 } }{ 2 } = 54.083 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 60**2+2 * 45**2 - 75**2 } }{ 2 } = 37.5 ; ;
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