7 9 12 triangle

Obtuse scalene triangle.

Sides: a = 7   b = 9   c = 12

Area: T = 31.3054951685
Perimeter: p = 28
Semiperimeter: s = 14

Angle ∠ A = α = 35.43109446873° = 35°25'51″ = 0.61883866419 rad
Angle ∠ B = β = 48.19896851042° = 48°11'23″ = 0.84110686706 rad
Angle ∠ C = γ = 96.37993702084° = 96°22'46″ = 1.68221373411 rad

Height: ha = 8.944427191
Height: hb = 6.957665593
Height: hc = 5.21774919475

Median: ma = 10.01224921973
Median: mb = 8.73221245983
Median: mc = 5.38551648071

Inradius: r = 2.23660679775
Circumradius: R = 6.03773835392

Vertex coordinates: A[12; 0] B[0; 0] C[4.66766666667; 5.21774919475]
Centroid: CG[5.55655555556; 1.73991639825]
Coordinates of the circumscribed circle: U[6; -0.67108203932]
Coordinates of the inscribed circle: I[5; 2.23660679775]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 144.5699055313° = 144°34'9″ = 0.61883866419 rad
∠ B' = β' = 131.8110314896° = 131°48'37″ = 0.84110686706 rad
∠ C' = γ' = 83.62106297916° = 83°37'14″ = 1.68221373411 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 = 12 ; ;

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

p = a+b+c = 7+9+12 = 28 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 28 }{ 2 } = 14 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 14 * (14-7)(14-9)(14-12) } ; ; T = sqrt{ 980 } = 31.3 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 31.3 }{ 7 } = 8.94 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 31.3 }{ 9 } = 6.96 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 31.3 }{ 12 } = 5.22 ; ;

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{ 7**2-9**2-12**2 }{ 2 * 9 * 12 } ) = 35° 25'51" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 9**2-7**2-12**2 }{ 2 * 7 * 12 } ) = 48° 11'23" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 12**2-7**2-9**2 }{ 2 * 9 * 7 } ) = 96° 22'46" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 31.3 }{ 14 } = 2.24 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 7 }{ 2 * sin 35° 25'51" } = 6.04 ; ;




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