9 24 28 triangle

Obtuse scalene triangle.

Sides: a = 9   b = 24   c = 28

Area: T = 103.2287600476
Perimeter: p = 61
Semiperimeter: s = 30.5

Angle ∠ A = α = 17.89220755157° = 17°53'31″ = 0.31222756278 rad
Angle ∠ B = β = 55.01114518867° = 55°41″ = 0.96601309617 rad
Angle ∠ C = γ = 107.0966472598° = 107°5'47″ = 1.86991860641 rad

Height: ha = 22.93994667724
Height: hb = 8.60223000397
Height: hc = 7.3733400034

Median: ma = 25.68655990781
Median: mb = 16.98552877515
Median: mc = 11.51108644332

Inradius: r = 3.3854511491
Circumradius: R = 14.64772454366

Vertex coordinates: A[28; 0] B[0; 0] C[5.16107142857; 7.3733400034]
Centroid: CG[11.05435714286; 2.45878000113]
Coordinates of the circumscribed circle: U[14; -4.30660189131]
Coordinates of the inscribed circle: I[6.5; 3.3854511491]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 162.1087924484° = 162°6'29″ = 0.31222756278 rad
∠ B' = β' = 124.9898548113° = 124°59'19″ = 0.96601309617 rad
∠ C' = γ' = 72.90435274025° = 72°54'13″ = 1.86991860641 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 = 24 ; ; c = 28 ; ;

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

p = a+b+c = 9+24+28 = 61 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 61 }{ 2 } = 30.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 30.5 * (30.5-9)(30.5-24)(30.5-28) } ; ; T = sqrt{ 10655.94 } = 103.23 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 103.23 }{ 9 } = 22.94 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 103.23 }{ 24 } = 8.6 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 103.23 }{ 28 } = 7.37 ; ;

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**2-24**2-28**2 }{ 2 * 24 * 28 } ) = 17° 53'31" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 24**2-9**2-28**2 }{ 2 * 9 * 28 } ) = 55° 41" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 28**2-9**2-24**2 }{ 2 * 24 * 9 } ) = 107° 5'47" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 103.23 }{ 30.5 } = 3.38 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9 }{ 2 * sin 17° 53'31" } = 14.65 ; ;




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