6 7 9 triangle

Acute scalene triangle.

Sides: a = 6   b = 7   c = 9

Area: T = 20.97661769634
Perimeter: p = 22
Semiperimeter: s = 11

Angle ∠ A = α = 41.7522205202° = 41°45'8″ = 0.72987134507 rad
Angle ∠ B = β = 50.97771974348° = 50°58'38″ = 0.89897199387 rad
Angle ∠ C = γ = 87.27105973632° = 87°16'14″ = 1.52331592642 rad

Height: ha = 6.99220589878
Height: hb = 5.99331934181
Height: hc = 4.66113726585

Median: ma = 7.48333147735
Median: mb = 6.80107352544
Median: mc = 4.7176990566

Inradius: r = 1.90769251785
Circumradius: R = 4.50551107342

Vertex coordinates: A[9; 0] B[0; 0] C[3.77877777778; 4.66113726585]
Centroid: CG[4.25992592593; 1.55437908862]
Coordinates of the circumscribed circle: U[4.5; 0.21545290826]
Coordinates of the inscribed circle: I[4; 1.90769251785]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 138.2487794798° = 138°14'52″ = 0.72987134507 rad
∠ B' = β' = 129.0232802565° = 129°1'22″ = 0.89897199387 rad
∠ C' = γ' = 92.72994026368° = 92°43'46″ = 1.52331592642 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 = 6 ; ; b = 7 ; ; c = 9 ; ;

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

p = a+b+c = 6+7+9 = 22 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 22 }{ 2 } = 11 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 11 * (11-6)(11-7)(11-9) } ; ; T = sqrt{ 440 } = 20.98 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 20.98 }{ 6 } = 6.99 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 20.98 }{ 7 } = 5.99 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 20.98 }{ 9 } = 4.66 ; ;

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{ 6**2-7**2-9**2 }{ 2 * 7 * 9 } ) = 41° 45'8" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 7**2-6**2-9**2 }{ 2 * 6 * 9 } ) = 50° 58'38" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 9**2-6**2-7**2 }{ 2 * 7 * 6 } ) = 87° 16'14" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 20.98 }{ 11 } = 1.91 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 6 }{ 2 * sin 41° 45'8" } = 4.51 ; ;




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