15 15 19 triangle

Acute isosceles triangle.

Sides: a = 15   b = 15   c = 19

Area: T = 110.2787774279
Perimeter: p = 49
Semiperimeter: s = 24.5

Angle ∠ A = α = 50.70435197608° = 50°42'13″ = 0.88549433622 rad
Angle ∠ B = β = 50.70435197608° = 50°42'13″ = 0.88549433622 rad
Angle ∠ C = γ = 78.59329604784° = 78°35'35″ = 1.37217059292 rad

Height: ha = 14.70437032372
Height: hb = 14.70437032372
Height: hc = 11.60881867662

Median: ma = 15.38766825534
Median: mb = 15.38766825534
Median: mc = 11.60881867662

Inradius: r = 4.50111336441
Circumradius: R = 9.69114360757

Vertex coordinates: A[19; 0] B[0; 0] C[9.5; 11.60881867662]
Centroid: CG[9.5; 3.86993955887]
Coordinates of the circumscribed circle: U[9.5; 1.91767506905]
Coordinates of the inscribed circle: I[9.5; 4.50111336441]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 129.2966480239° = 129°17'47″ = 0.88549433622 rad
∠ B' = β' = 129.2966480239° = 129°17'47″ = 0.88549433622 rad
∠ C' = γ' = 101.4077039522° = 101°24'25″ = 1.37217059292 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 = 15 ; ; b = 15 ; ; c = 19 ; ;

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

p = a+b+c = 15+15+19 = 49 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 49 }{ 2 } = 24.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 24.5 * (24.5-15)(24.5-15)(24.5-19) } ; ; T = sqrt{ 12161.19 } = 110.28 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 110.28 }{ 15 } = 14.7 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 110.28 }{ 15 } = 14.7 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 110.28 }{ 19 } = 11.61 ; ;

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{ 15**2-15**2-19**2 }{ 2 * 15 * 19 } ) = 50° 42'13" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 15**2-15**2-19**2 }{ 2 * 15 * 19 } ) = 50° 42'13" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 19**2-15**2-15**2 }{ 2 * 15 * 15 } ) = 78° 35'35" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 110.28 }{ 24.5 } = 4.5 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 15 }{ 2 * sin 50° 42'13" } = 9.69 ; ;




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