78 110 90 triangle

Acute scalene triangle.

Sides: a = 78   b = 110   c = 90

Area: T = 3471.118783148
Perimeter: p = 278
Semiperimeter: s = 139

Angle ∠ A = α = 44.52662467768° = 44°31'34″ = 0.77771296098 rad
Angle ∠ B = β = 81.46438696679° = 81°27'50″ = 1.42218127471 rad
Angle ∠ C = γ = 54.01098835553° = 54°36″ = 0.94326502967 rad

Height: ha = 89.00330213199
Height: hb = 63.11112332996
Height: hc = 77.13659518106

Median: ma = 92.6232891339
Median: mb = 63.7733035054
Median: mc = 84.06554506917

Inradius: r = 24.97220707301
Circumradius: R = 55.61660895056

Vertex coordinates: A[90; 0] B[0; 0] C[11.57877777778; 77.13659518106]
Centroid: CG[33.85992592593; 25.71219839369]
Coordinates of the circumscribed circle: U[45; 32.68325551617]
Coordinates of the inscribed circle: I[29; 24.97220707301]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135.4743753223° = 135°28'26″ = 0.77771296098 rad
∠ B' = β' = 98.53661303321° = 98°32'10″ = 1.42218127471 rad
∠ C' = γ' = 125.9990116445° = 125°59'24″ = 0.94326502967 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 = 78 ; ; b = 110 ; ; c = 90 ; ;

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

p = a+b+c = 78+110+90 = 278 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 278 }{ 2 } = 139 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 139 * (139-78)(139-110)(139-90) } ; ; T = sqrt{ 12048659 } = 3471.12 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3471.12 }{ 78 } = 89 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3471.12 }{ 110 } = 63.11 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3471.12 }{ 90 } = 77.14 ; ;

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{ 110**2+90**2-78**2 }{ 2 * 110 * 90 } ) = 44° 31'34" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 78**2+90**2-110**2 }{ 2 * 78 * 90 } ) = 81° 27'50" ; ;
 gamma = 180° - alpha - beta = 180° - 44° 31'34" - 81° 27'50" = 54° 36" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3471.12 }{ 139 } = 24.97 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 78 }{ 2 * sin 44° 31'34" } = 55.62 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 110**2+2 * 90**2 - 78**2 } }{ 2 } = 92.623 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 78**2 - 110**2 } }{ 2 } = 63.773 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 110**2+2 * 78**2 - 90**2 } }{ 2 } = 84.065 ; ;
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