65 72 97 triangle

Right scalene triangle.

Sides: a = 65   b = 72   c = 97

Area: T = 2340
Perimeter: p = 234
Semiperimeter: s = 117

Angle ∠ A = α = 42.07550220508° = 42°4'30″ = 0.73443476676 rad
Angle ∠ B = β = 47.92549779492° = 47°55'30″ = 0.83664486592 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 72
Height: hb = 65
Height: hc = 48.24774226804

Median: ma = 78.99552530219
Median: mb = 74.3033431953
Median: mc = 48.5

Inradius: r = 20
Circumradius: R = 48.5

Vertex coordinates: A[97; 0] B[0; 0] C[43.55767010309; 48.24774226804]
Centroid: CG[46.8522233677; 16.08224742268]
Coordinates of the circumscribed circle: U[48.5; 0]
Coordinates of the inscribed circle: I[45; 20]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 137.9254977949° = 137°55'30″ = 0.73443476676 rad
∠ B' = β' = 132.0755022051° = 132°4'30″ = 0.83664486592 rad
∠ C' = γ' = 90° = 1.57107963268 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 = 65 ; ; b = 72 ; ; c = 97 ; ;

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

p = a+b+c = 65+72+97 = 234 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 234 }{ 2 } = 117 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 117 * (117-65)(117-72)(117-97) } ; ; T = sqrt{ 5475600 } = 2340 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2340 }{ 65 } = 72 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2340 }{ 72 } = 65 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2340 }{ 97 } = 48.25 ; ;

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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 65**2-72**2-97**2 }{ 2 * 72 * 97 } ) = 42° 4'30" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 72**2-65**2-97**2 }{ 2 * 65 * 97 } ) = 47° 55'30" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 97**2-65**2-72**2 }{ 2 * 72 * 65 } ) = 90° ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2340 }{ 117 } = 20 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 65 }{ 2 * sin 42° 4'30" } = 48.5 ; ;

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