39 52 65 triangle

Right scalene Pythagorean triangle.

Sides: a = 39   b = 52   c = 65

Area: T = 1014
Perimeter: p = 156
Semiperimeter: s = 78

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 = 52
Height: hb = 39
Height: hc = 31.2

Median: ma = 55.53660243446
Median: mb = 46.8722166581
Median: mc = 32.5

Inradius: r = 13
Circumradius: R = 32.5

Vertex coordinates: A[65; 0] B[0; 0] C[23.4; 31.2]
Centroid: CG[29.46766666667; 10.4]
Coordinates of the circumscribed circle: U[32.5; 0]
Coordinates of the inscribed circle: I[26; 13]

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?

a = 39 ; ; b = 52 ; ; c = 65 ; ;

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

p = a+b+c = 39+52+65 = 156 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 156 }{ 2 } = 78 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 78 * (78-39)(78-52)(78-65) } ; ; T = sqrt{ 1028196 } = 1014 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1014 }{ 39 } = 52 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1014 }{ 52 } = 39 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1014 }{ 65 } = 31.2 ; ;

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{ 52**2+65**2-39**2 }{ 2 * 52 * 65 } ) = 36° 52'12" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 39**2+65**2-52**2 }{ 2 * 39 * 65 } ) = 53° 7'48" ; ; gamma = 180° - alpha - beta = 180° - 36° 52'12" - 53° 7'48" = 90° ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1014 }{ 78 } = 13 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 39 }{ 2 * sin 36° 52'12" } = 32.5 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 52**2+2 * 65**2 - 39**2 } }{ 2 } = 55.536 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 65**2+2 * 39**2 - 52**2 } }{ 2 } = 46.872 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 52**2+2 * 39**2 - 65**2 } }{ 2 } = 32.5 ; ;
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