7 9 10 triangle

Acute scalene triangle.

Sides: a = 7   b = 9   c = 10

Area: T = 30.59441170816
Perimeter: p = 26
Semiperimeter: s = 13

Angle ∠ A = α = 42.83334280661° = 42°50' = 0.74875843497 rad
Angle ∠ B = β = 60.9410718932° = 60°56'27″ = 1.06436161939 rad
Angle ∠ C = γ = 76.2265853002° = 76°13'33″ = 1.333039211 rad

Height: ha = 8.7411176309
Height: hb = 6.79986926848
Height: hc = 6.11988234163

Median: ma = 8.84659030065
Median: mb = 7.36554599313
Median: mc = 6.32545553203

Inradius: r = 2.35333936217
Circumradius: R = 5.14880485474

Vertex coordinates: A[10; 0] B[0; 0] C[3.4; 6.11988234163]
Centroid: CG[4.46766666667; 2.04396078054]
Coordinates of the circumscribed circle: U[5; 1.22657258446]
Coordinates of the inscribed circle: I[4; 2.35333936217]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 137.1676571934° = 137°10' = 0.74875843497 rad
∠ B' = β' = 119.0599281068° = 119°3'33″ = 1.06436161939 rad
∠ C' = γ' = 103.7744146998° = 103°46'27″ = 1.333039211 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 = 7 ; ; b = 9 ; ; c = 10 ; ;

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

p = a+b+c = 7+9+10 = 26 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 26 }{ 2 } = 13 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 13 * (13-7)(13-9)(13-10) } ; ; T = sqrt{ 936 } = 30.59 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 30.59 }{ 7 } = 8.74 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 30.59 }{ 9 } = 6.8 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 30.59 }{ 10 } = 6.12 ; ;

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{ 7**2-9**2-10**2 }{ 2 * 9 * 10 } ) = 42° 50' ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 9**2-7**2-10**2 }{ 2 * 7 * 10 } ) = 60° 56'27" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 10**2-7**2-9**2 }{ 2 * 9 * 7 } ) = 76° 13'33" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 30.59 }{ 13 } = 2.35 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 7 }{ 2 * sin 42° 50' } = 5.15 ; ;




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