8 9 10 triangle

Acute scalene triangle.

Sides: a = 8   b = 9   c = 10

Area: T = 34.19770393455
Perimeter: p = 27
Semiperimeter: s = 13.5

Angle ∠ A = α = 49.45883981265° = 49°27'30″ = 0.86332118901 rad
Angle ∠ B = β = 58.75215587378° = 58°45'6″ = 1.02554081407 rad
Angle ∠ C = γ = 71.79900431357° = 71°47'24″ = 1.25329726229 rad

Height: ha = 8.54992598364
Height: hb = 7.59993420768
Height: hc = 6.83994078691

Median: ma = 8.63113382508
Median: mb = 7.85881168228
Median: mc = 6.8922024376

Inradius: r = 2.53331140256
Circumradius: R = 5.26436135597

Vertex coordinates: A[10; 0] B[0; 0] C[4.15; 6.83994078691]
Centroid: CG[4.71766666667; 2.2879802623]
Coordinates of the circumscribed circle: U[5; 1.64548792374]
Coordinates of the inscribed circle: I[4.5; 2.53331140256]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 130.5421601874° = 130°32'30″ = 0.86332118901 rad
∠ B' = β' = 121.2488441262° = 121°14'54″ = 1.02554081407 rad
∠ C' = γ' = 108.2109956864° = 108°12'36″ = 1.25329726229 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 = 8 ; ; b = 9 ; ; c = 10 ; ;

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

p = a+b+c = 8+9+10 = 27 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 27 }{ 2 } = 13.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 13.5 * (13.5-8)(13.5-9)(13.5-10) } ; ; T = sqrt{ 1169.44 } = 34.2 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 34.2 }{ 8 } = 8.55 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 34.2 }{ 9 } = 7.6 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 34.2 }{ 10 } = 6.84 ; ;

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{ 8**2-9**2-10**2 }{ 2 * 9 * 10 } ) = 49° 27'30" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 9**2-8**2-10**2 }{ 2 * 8 * 10 } ) = 58° 45'6" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 10**2-8**2-9**2 }{ 2 * 9 * 8 } ) = 71° 47'24" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 34.2 }{ 13.5 } = 2.53 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8 }{ 2 * sin 49° 27'30" } = 5.26 ; ;




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