3 9 10 triangle

Obtuse scalene triangle.

Sides: a = 3   b = 9   c = 10

Area: T = 13.26664991614
Perimeter: p = 22
Semiperimeter: s = 11

Angle ∠ A = α = 17.14662099989° = 17°8'46″ = 0.29992578187 rad
Angle ∠ B = β = 62.18218607153° = 62°10'55″ = 1.08552782045 rad
Angle ∠ C = γ = 100.6721929286° = 100°40'19″ = 1.75770566304 rad

Height: ha = 8.84443327743
Height: hb = 2.94881109248
Height: hc = 2.65332998323

Median: ma = 9.3944147114
Median: mb = 5.85223499554
Median: mc = 4.4722135955

Inradius: r = 1.20660453783
Circumradius: R = 5.08880039397

Vertex coordinates: A[10; 0] B[0; 0] C[1.4; 2.65332998323]
Centroid: CG[3.8; 0.88444332774]
Coordinates of the circumscribed circle: U[5; -0.94222229518]
Coordinates of the inscribed circle: I[2; 1.20660453783]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 162.8543790001° = 162°51'14″ = 0.29992578187 rad
∠ B' = β' = 117.8188139285° = 117°49'5″ = 1.08552782045 rad
∠ C' = γ' = 79.32880707142° = 79°19'41″ = 1.75770566304 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 = 3 ; ; b = 9 ; ; c = 10 ; ;

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

p = a+b+c = 3+9+10 = 22 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 22 }{ 2 } = 11 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 11 * (11-3)(11-9)(11-10) } ; ; T = sqrt{ 176 } = 13.27 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 13.27 }{ 3 } = 8.84 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 13.27 }{ 9 } = 2.95 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 13.27 }{ 10 } = 2.65 ; ;

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{ 3**2-9**2-10**2 }{ 2 * 9 * 10 } ) = 17° 8'46" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 9**2-3**2-10**2 }{ 2 * 3 * 10 } ) = 62° 10'55" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 10**2-3**2-9**2 }{ 2 * 9 * 3 } ) = 100° 40'19" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 13.27 }{ 11 } = 1.21 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 3 }{ 2 * sin 17° 8'46" } = 5.09 ; ;




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