8 9 9 triangle

Acute isosceles triangle.

Sides: a = 8   b = 9   c = 9

Area: T = 32.24990309932
Perimeter: p = 26
Semiperimeter: s = 13

Angle ∠ A = α = 52.77655999225° = 52°46'32″ = 0.92111079834 rad
Angle ∠ B = β = 63.61222000388° = 63°36'44″ = 1.11102423351 rad
Angle ∠ C = γ = 63.61222000388° = 63°36'44″ = 1.11102423351 rad

Height: ha = 8.06222577483
Height: hb = 7.16664513318
Height: hc = 7.16664513318

Median: ma = 8.06222577483
Median: mb = 7.22884161474
Median: mc = 7.22884161474

Inradius: r = 2.48106946918
Circumradius: R = 5.02334067509

Vertex coordinates: A[9; 0] B[0; 0] C[3.55655555556; 7.16664513318]
Centroid: CG[4.18551851852; 2.38988171106]
Coordinates of the circumscribed circle: U[4.5; 2.23326252226]
Coordinates of the inscribed circle: I[4; 2.48106946918]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 127.2244400078° = 127°13'28″ = 0.92111079834 rad
∠ B' = β' = 116.3887799961° = 116°23'16″ = 1.11102423351 rad
∠ C' = γ' = 116.3887799961° = 116°23'16″ = 1.11102423351 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 = 8 ; ; b = 9 ; ; c = 9 ; ;

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

p = a+b+c = 8+9+9 = 26 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 26 }{ 2 } = 13 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 13 * (13-8)(13-9)(13-9) } ; ; T = sqrt{ 1040 } = 32.25 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 32.25 }{ 8 } = 8.06 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 32.25 }{ 9 } = 7.17 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 32.25 }{ 9 } = 7.17 ; ;

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{ 8**2-9**2-9**2 }{ 2 * 9 * 9 } ) = 52° 46'32" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 9**2-8**2-9**2 }{ 2 * 8 * 9 } ) = 63° 36'44" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 9**2-8**2-9**2 }{ 2 * 9 * 8 } ) = 63° 36'44" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 32.25 }{ 13 } = 2.48 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8 }{ 2 * sin 52° 46'32" } = 5.02 ; ;




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