9 12 14 triangle

Acute scalene triangle.

Sides: a = 9   b = 12   c = 14

Area: T = 53.511109698
Perimeter: p = 35
Semiperimeter: s = 17.5

Angle ∠ A = α = 39.57112194572° = 39°34'16″ = 0.69106480686 rad
Angle ∠ B = β = 58.1454569176° = 58°8'40″ = 1.01548141743 rad
Angle ∠ C = γ = 82.28442113668° = 82°17'3″ = 1.43661304108 rad

Height: ha = 11.89113548844
Height: hb = 8.91985161633
Height: hc = 7.64444424257

Median: ma = 12.23772382505
Median: mb = 10.12442283657
Median: mc = 7.96986887253

Inradius: r = 3.05877769703
Circumradius: R = 7.06439553538

Vertex coordinates: A[14; 0] B[0; 0] C[4.75; 7.64444424257]
Centroid: CG[6.25; 2.54881474752]
Coordinates of the circumscribed circle: U[7; 0.94884014132]
Coordinates of the inscribed circle: I[5.5; 3.05877769703]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 140.4298780543° = 140°25'44″ = 0.69106480686 rad
∠ B' = β' = 121.8555430824° = 121°51'20″ = 1.01548141743 rad
∠ C' = γ' = 97.71657886332° = 97°42'57″ = 1.43661304108 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 = 12 ; ; c = 14 ; ;

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

p = a+b+c = 9+12+14 = 35 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 35 }{ 2 } = 17.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 17.5 * (17.5-9)(17.5-12)(17.5-14) } ; ; T = sqrt{ 2863.44 } = 53.51 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 53.51 }{ 9 } = 11.89 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 53.51 }{ 12 } = 8.92 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 53.51 }{ 14 } = 7.64 ; ;

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**2-12**2-14**2 }{ 2 * 12 * 14 } ) = 39° 34'16" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 12**2-9**2-14**2 }{ 2 * 9 * 14 } ) = 58° 8'40" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 14**2-9**2-12**2 }{ 2 * 12 * 9 } ) = 82° 17'3" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 53.51 }{ 17.5 } = 3.06 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9 }{ 2 * sin 39° 34'16" } = 7.06 ; ;




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