9 9 12 triangle

Acute isosceles triangle.

Sides: a = 9   b = 9   c = 12

Area: T = 40.2499223595
Perimeter: p = 30
Semiperimeter: s = 15

Angle ∠ A = α = 48.19896851042° = 48°11'23″ = 0.84110686706 rad
Angle ∠ B = β = 48.19896851042° = 48°11'23″ = 0.84110686706 rad
Angle ∠ C = γ = 83.62106297916° = 83°37'14″ = 1.45994553125 rad

Height: ha = 8.944427191
Height: hb = 8.944427191
Height: hc = 6.70882039325

Median: ma = 9.60546863561
Median: mb = 9.60546863561
Median: mc = 6.70882039325

Inradius: r = 2.6833281573
Circumradius: R = 6.03773835392

Vertex coordinates: A[12; 0] B[0; 0] C[6; 6.70882039325]
Centroid: CG[6; 2.23660679775]
Coordinates of the circumscribed circle: U[6; 0.67108203932]
Coordinates of the inscribed circle: I[6; 2.6833281573]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 131.8110314896° = 131°48'37″ = 0.84110686706 rad
∠ B' = β' = 131.8110314896° = 131°48'37″ = 0.84110686706 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 = 9 ; ; b = 9 ; ; c = 12 ; ;

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

p = a+b+c = 9+9+12 = 30 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 30 }{ 2 } = 15 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 15 * (15-9)(15-9)(15-12) } ; ; T = sqrt{ 1620 } = 40.25 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 40.25 }{ 9 } = 8.94 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 40.25 }{ 9 } = 8.94 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 40.25 }{ 12 } = 6.71 ; ;

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**2-9**2-12**2 }{ 2 * 9 * 12 } ) = 48° 11'23" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 9**2-9**2-12**2 }{ 2 * 9 * 12 } ) = 48° 11'23" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 12**2-9**2-9**2 }{ 2 * 9 * 9 } ) = 83° 37'14" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 40.25 }{ 15 } = 2.68 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9 }{ 2 * sin 48° 11'23" } = 6.04 ; ;




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