9 13 13 triangle

Acute isosceles triangle.

Sides: a = 9   b = 13   c = 13

Area: T = 54.88333991294
Perimeter: p = 35
Semiperimeter: s = 17.5

Angle ∠ A = α = 40.50444934844° = 40°30'16″ = 0.70769367732 rad
Angle ∠ B = β = 69.74877532578° = 69°44'52″ = 1.21773279402 rad
Angle ∠ C = γ = 69.74877532578° = 69°44'52″ = 1.21773279402 rad

Height: ha = 12.19663109177
Height: hb = 8.44435998661
Height: hc = 8.44435998661

Median: ma = 12.19663109177
Median: mb = 9.09767026993
Median: mc = 9.09767026993

Inradius: r = 3.1366194236
Circumradius: R = 6.92883245213

Vertex coordinates: A[13; 0] B[0; 0] C[3.11553846154; 8.44435998661]
Centroid: CG[5.37217948718; 2.81545332887]
Coordinates of the circumscribed circle: U[6.5; 2.39882661804]
Coordinates of the inscribed circle: I[4.5; 3.1366194236]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 139.4965506516° = 139°29'44″ = 0.70769367732 rad
∠ B' = β' = 110.2522246742° = 110°15'8″ = 1.21773279402 rad
∠ C' = γ' = 110.2522246742° = 110°15'8″ = 1.21773279402 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 = 13 ; ; c = 13 ; ;

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

p = a+b+c = 9+13+13 = 35 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 35 }{ 2 } = 17.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 17.5 * (17.5-9)(17.5-13)(17.5-13) } ; ; T = sqrt{ 3012.19 } = 54.88 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 54.88 }{ 9 } = 12.2 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 54.88 }{ 13 } = 8.44 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 54.88 }{ 13 } = 8.44 ; ;

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**2-13**2-13**2 }{ 2 * 13 * 13 } ) = 40° 30'16" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 13**2-9**2-13**2 }{ 2 * 9 * 13 } ) = 69° 44'52" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 13**2-9**2-13**2 }{ 2 * 13 * 9 } ) = 69° 44'52" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 54.88 }{ 17.5 } = 3.14 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9 }{ 2 * sin 40° 30'16" } = 6.93 ; ;




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