9 10 10 triangle

Acute isosceles triangle.

Sides: a = 9   b = 10   c = 10

Area: T = 40.18662849739
Perimeter: p = 29
Semiperimeter: s = 14.5

Angle ∠ A = α = 53.48773679008° = 53°29'15″ = 0.93435306781 rad
Angle ∠ B = β = 63.25663160496° = 63°15'23″ = 1.10440309877 rad
Angle ∠ C = γ = 63.25663160496° = 63°15'23″ = 1.10440309877 rad

Height: ha = 8.93302855497
Height: hb = 8.03772569948
Height: hc = 8.03772569948

Median: ma = 8.93302855497
Median: mb = 8.09332070281
Median: mc = 8.09332070281

Inradius: r = 2.77114679292
Circumradius: R = 5.59989251096

Vertex coordinates: A[10; 0] B[0; 0] C[4.05; 8.03772569948]
Centroid: CG[4.68333333333; 2.67990856649]
Coordinates of the circumscribed circle: U[5; 2.52195162993]
Coordinates of the inscribed circle: I[4.5; 2.77114679292]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 126.5132632099° = 126°30'45″ = 0.93435306781 rad
∠ B' = β' = 116.744368395° = 116°44'37″ = 1.10440309877 rad
∠ C' = γ' = 116.744368395° = 116°44'37″ = 1.10440309877 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 = 10 ; ; c = 10 ; ;

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

p = a+b+c = 9+10+10 = 29 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 29 }{ 2 } = 14.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 14.5 * (14.5-9)(14.5-10)(14.5-10) } ; ; T = sqrt{ 1614.94 } = 40.19 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 40.19 }{ 9 } = 8.93 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 40.19 }{ 10 } = 8.04 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 40.19 }{ 10 } = 8.04 ; ;

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**2-10**2-10**2 }{ 2 * 10 * 10 } ) = 53° 29'15" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 10**2-9**2-10**2 }{ 2 * 9 * 10 } ) = 63° 15'23" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 10**2-9**2-10**2 }{ 2 * 10 * 9 } ) = 63° 15'23" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 40.19 }{ 14.5 } = 2.77 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9 }{ 2 * sin 53° 29'15" } = 5.6 ; ;




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