9 11 11 triangle

Acute isosceles triangle.

Sides: a = 9   b = 11   c = 11

Area: T = 45.16884347747
Perimeter: p = 31
Semiperimeter: s = 15.5

Angle ∠ A = α = 48.2955479834° = 48°17'44″ = 0.84329151369 rad
Angle ∠ B = β = 65.8522260083° = 65°51'8″ = 1.14993387583 rad
Angle ∠ C = γ = 65.8522260083° = 65°51'8″ = 1.14993387583 rad

Height: ha = 10.03774299499
Height: hb = 8.21224426863
Height: hc = 8.21224426863

Median: ma = 10.03774299499
Median: mb = 8.41113019206
Median: mc = 8.41113019206

Inradius: r = 2.91440925661
Circumradius: R = 6.02774393248

Vertex coordinates: A[11; 0] B[0; 0] C[3.68218181818; 8.21224426863]
Centroid: CG[4.89439393939; 2.73774808954]
Coordinates of the circumscribed circle: U[5.5; 2.46657706329]
Coordinates of the inscribed circle: I[4.5; 2.91440925661]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 131.7054520166° = 131°42'16″ = 0.84329151369 rad
∠ B' = β' = 114.1487739917° = 114°8'52″ = 1.14993387583 rad
∠ C' = γ' = 114.1487739917° = 114°8'52″ = 1.14993387583 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 = 11 ; ; c = 11 ; ;

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

p = a+b+c = 9+11+11 = 31 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 31 }{ 2 } = 15.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 15.5 * (15.5-9)(15.5-11)(15.5-11) } ; ; T = sqrt{ 2040.19 } = 45.17 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 45.17 }{ 9 } = 10.04 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 45.17 }{ 11 } = 8.21 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 45.17 }{ 11 } = 8.21 ; ;

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**2-11**2-11**2 }{ 2 * 11 * 11 } ) = 48° 17'44" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 11**2-9**2-11**2 }{ 2 * 9 * 11 } ) = 65° 51'8" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 11**2-9**2-11**2 }{ 2 * 11 * 9 } ) = 65° 51'8" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 45.17 }{ 15.5 } = 2.91 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9 }{ 2 * sin 48° 17'44" } = 6.03 ; ;




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