5 9 10 triangle

Acute scalene triangle.

Sides: a = 5   b = 9   c = 10

Area: T = 22.45499443206
Perimeter: p = 24
Semiperimeter: s = 12

Angle ∠ A = α = 29.92664348666° = 29°55'35″ = 0.52223148218 rad
Angle ∠ B = β = 63.89661188627° = 63°53'46″ = 1.11551976534 rad
Angle ∠ C = γ = 86.17774462707° = 86°10'39″ = 1.50440801784 rad

Height: ha = 8.98799777283
Height: hb = 4.98988765157
Height: hc = 4.49899888641

Median: ma = 9.17987798753
Median: mb = 6.5
Median: mc = 5.29215026221

Inradius: r = 1.87108286934
Circumradius: R = 5.01111482859

Vertex coordinates: A[10; 0] B[0; 0] C[2.2; 4.49899888641]
Centroid: CG[4.06766666667; 1.49766629547]
Coordinates of the circumscribed circle: U[5; 0.33440765524]
Coordinates of the inscribed circle: I[3; 1.87108286934]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 150.0743565133° = 150°4'25″ = 0.52223148218 rad
∠ B' = β' = 116.1043881137° = 116°6'14″ = 1.11551976534 rad
∠ C' = γ' = 93.82325537293° = 93°49'21″ = 1.50440801784 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 = 5 ; ; b = 9 ; ; c = 10 ; ;

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

p = a+b+c = 5+9+10 = 24 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 24 }{ 2 } = 12 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 12 * (12-5)(12-9)(12-10) } ; ; T = sqrt{ 504 } = 22.45 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 22.45 }{ 5 } = 8.98 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 22.45 }{ 9 } = 4.99 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 22.45 }{ 10 } = 4.49 ; ;

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{ 5**2-9**2-10**2 }{ 2 * 9 * 10 } ) = 29° 55'35" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 9**2-5**2-10**2 }{ 2 * 5 * 10 } ) = 63° 53'46" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 10**2-5**2-9**2 }{ 2 * 9 * 5 } ) = 86° 10'39" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 22.45 }{ 12 } = 1.87 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 5 }{ 2 * sin 29° 55'35" } = 5.01 ; ;




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