3 10 10 triangle

Acute isosceles triangle.

Sides: a = 3   b = 10   c = 10

Area: T = 14.833028995
Perimeter: p = 23
Semiperimeter: s = 11.5

Angle ∠ A = α = 17.25438531174° = 17°15'14″ = 0.30111365456 rad
Angle ∠ B = β = 81.37330734413° = 81°22'23″ = 1.4220228054 rad
Angle ∠ C = γ = 81.37330734413° = 81°22'23″ = 1.4220228054 rad

Height: ha = 9.88768599666
Height: hb = 2.966605799
Height: hc = 2.966605799

Median: ma = 9.88768599666
Median: mb = 5.43113902456
Median: mc = 5.43113902456

Inradius: r = 1.29895904304
Circumradius: R = 5.05772173742

Vertex coordinates: A[10; 0] B[0; 0] C[0.45; 2.966605799]
Centroid: CG[3.48333333333; 0.98986859967]
Coordinates of the circumscribed circle: U[5; 0.75985826061]
Coordinates of the inscribed circle: I[1.5; 1.29895904304]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 162.7466146883° = 162°44'46″ = 0.30111365456 rad
∠ B' = β' = 98.62769265587° = 98°37'37″ = 1.4220228054 rad
∠ C' = γ' = 98.62769265587° = 98°37'37″ = 1.4220228054 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 = 3 ; ; b = 10 ; ; c = 10 ; ;

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

p = a+b+c = 3+10+10 = 23 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 23 }{ 2 } = 11.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 11.5 * (11.5-3)(11.5-10)(11.5-10) } ; ; T = sqrt{ 219.94 } = 14.83 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 14.83 }{ 3 } = 9.89 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 14.83 }{ 10 } = 2.97 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 14.83 }{ 10 } = 2.97 ; ;

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{ 3**2-10**2-10**2 }{ 2 * 10 * 10 } ) = 17° 15'14" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 10**2-3**2-10**2 }{ 2 * 3 * 10 } ) = 81° 22'23" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 10**2-3**2-10**2 }{ 2 * 10 * 3 } ) = 81° 22'23" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 14.83 }{ 11.5 } = 1.29 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 3 }{ 2 * sin 17° 15'14" } = 5.06 ; ;




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