3 9 9 triangle

Acute isosceles triangle.

Sides: a = 3   b = 9   c = 9

Area: T = 13.3111179512
Perimeter: p = 21
Semiperimeter: s = 10.5

Angle ∠ A = α = 19.18881364537° = 19°11'17″ = 0.33548961584 rad
Angle ∠ B = β = 80.40659317731° = 80°24'21″ = 1.40333482476 rad
Angle ∠ C = γ = 80.40659317731° = 80°24'21″ = 1.40333482476 rad

Height: ha = 8.87441196746
Height: hb = 2.95880398915
Height: hc = 2.95880398915

Median: ma = 8.87441196746
Median: mb = 4.97549371855
Median: mc = 4.97549371855

Inradius: r = 1.26877313821
Circumradius: R = 4.56438329755

Vertex coordinates: A[9; 0] B[0; 0] C[0.5; 2.95880398915]
Centroid: CG[3.16766666667; 0.98660132972]
Coordinates of the circumscribed circle: U[4.5; 0.76106388293]
Coordinates of the inscribed circle: I[1.5; 1.26877313821]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 160.8121863546° = 160°48'43″ = 0.33548961584 rad
∠ B' = β' = 99.59440682269° = 99°35'39″ = 1.40333482476 rad
∠ C' = γ' = 99.59440682269° = 99°35'39″ = 1.40333482476 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 = 3 ; ; b = 9 ; ; c = 9 ; ;

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

p = a+b+c = 3+9+9 = 21 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 21 }{ 2 } = 10.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 10.5 * (10.5-3)(10.5-9)(10.5-9) } ; ; T = sqrt{ 177.19 } = 13.31 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 13.31 }{ 3 } = 8.87 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 13.31 }{ 9 } = 2.96 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 13.31 }{ 9 } = 2.96 ; ;

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{ 3**2-9**2-9**2 }{ 2 * 9 * 9 } ) = 19° 11'17" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 9**2-3**2-9**2 }{ 2 * 3 * 9 } ) = 80° 24'21" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 9**2-3**2-9**2 }{ 2 * 9 * 3 } ) = 80° 24'21" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 13.31 }{ 10.5 } = 1.27 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 3 }{ 2 * sin 19° 11'17" } = 4.56 ; ;




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