2 9 9 triangle

Acute isosceles triangle.

Sides: a = 2   b = 9   c = 9

Area: T = 8.944427191
Perimeter: p = 20
Semiperimeter: s = 10

Angle ∠ A = α = 12.75987404169° = 12°45'31″ = 0.22326820287 rad
Angle ∠ B = β = 83.62106297916° = 83°37'14″ = 1.45994553125 rad
Angle ∠ C = γ = 83.62106297916° = 83°37'14″ = 1.45994553125 rad

Height: ha = 8.944427191
Height: hb = 1.988761598
Height: hc = 1.988761598

Median: ma = 8.944427191
Median: mb = 4.7176990566
Median: mc = 4.7176990566

Inradius: r = 0.8944427191
Circumradius: R = 4.52880376544

Vertex coordinates: A[9; 0] B[0; 0] C[0.22222222222; 1.988761598]
Centroid: CG[3.07440740741; 0.663253866]
Coordinates of the circumscribed circle: U[4.5; 0.50331152949]
Coordinates of the inscribed circle: I[1; 0.8944427191]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 167.2411259583° = 167°14'29″ = 0.22326820287 rad
∠ B' = β' = 96.37993702084° = 96°22'46″ = 1.45994553125 rad
∠ C' = γ' = 96.37993702084° = 96°22'46″ = 1.45994553125 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 = 2 ; ; b = 9 ; ; c = 9 ; ;

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

p = a+b+c = 2+9+9 = 20 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 20 }{ 2 } = 10 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 10 * (10-2)(10-9)(10-9) } ; ; T = sqrt{ 80 } = 8.94 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 8.94 }{ 2 } = 8.94 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 8.94 }{ 9 } = 1.99 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 8.94 }{ 9 } = 1.99 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 9**2+9**2-2**2 }{ 2 * 9 * 9 } ) = 12° 45'31" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 2**2+9**2-9**2 }{ 2 * 2 * 9 } ) = 83° 37'14" ; ; gamma = 180° - alpha - beta = 180° - 12° 45'31" - 83° 37'14" = 83° 37'14" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 8.94 }{ 10 } = 0.89 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 2 }{ 2 * sin 12° 45'31" } = 4.53 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 9**2+2 * 9**2 - 2**2 } }{ 2 } = 8.944 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 9**2+2 * 2**2 - 9**2 } }{ 2 } = 4.717 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 9**2+2 * 2**2 - 9**2 } }{ 2 } = 4.717 ; ;
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