9 14 15 triangle

Acute scalene triangle.

Sides: a = 9   b = 14   c = 15

Area: T = 61.64441400297
Perimeter: p = 38
Semiperimeter: s = 19

Angle ∠ A = α = 35.95105676196° = 35°57'2″ = 0.62774557729 rad
Angle ∠ B = β = 65.95879240942° = 65°57'29″ = 1.15111829432 rad
Angle ∠ C = γ = 78.09215082861° = 78°5'29″ = 1.36329539374 rad

Height: ha = 13.69986977844
Height: hb = 8.80663057185
Height: hc = 8.21992186706

Median: ma = 13.79331142241
Median: mb = 10.19880390272
Median: mc = 9.06991785736

Inradius: r = 3.24444284226
Circumradius: R = 7.66549621484

Vertex coordinates: A[15; 0] B[0; 0] C[3.66766666667; 8.21992186706]
Centroid: CG[6.22222222222; 2.74397395569]
Coordinates of the circumscribed circle: U[7.5; 1.5821658856]
Coordinates of the inscribed circle: I[5; 3.24444284226]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 144.049943238° = 144°2'58″ = 0.62774557729 rad
∠ B' = β' = 114.0422075906° = 114°2'31″ = 1.15111829432 rad
∠ C' = γ' = 101.9088491714° = 101°54'31″ = 1.36329539374 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 = 9 ; ; b = 14 ; ; c = 15 ; ;

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

p = a+b+c = 9+14+15 = 38 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 38 }{ 2 } = 19 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 19 * (19-9)(19-14)(19-15) } ; ; T = sqrt{ 3800 } = 61.64 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 61.64 }{ 9 } = 13.7 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 61.64 }{ 14 } = 8.81 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 61.64 }{ 15 } = 8.22 ; ;

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{ 9**2-14**2-15**2 }{ 2 * 14 * 15 } ) = 35° 57'2" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 14**2-9**2-15**2 }{ 2 * 9 * 15 } ) = 65° 57'29" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 15**2-9**2-14**2 }{ 2 * 14 * 9 } ) = 78° 5'29" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 61.64 }{ 19 } = 3.24 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9 }{ 2 * sin 35° 57'2" } = 7.66 ; ;




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