in a triangle, A=32,C=65 find the value of x, y.
Answers
YOU CAN DO LIKE THIS a = 33 ; ; b = 56 ; ; c = 65 ; ;
1. The triangle circumference is the sum of the lengths of its three sides
p = a+b+c = 33+56+65 = 154 ; ;
2. Semiperimeter of the triangle
s = fraction{ o }{ 2 } = fraction{ 154 }{ 2 } = 77 ; ;
3. The triangle area using Heron's formula
T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 77 * (77-33)(77-56)(77-65) } ; ; T = sqrt{ 853776 } = 924 ; ;
4. Calculate the heights of the triangle from its area.
T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 924 }{ 33 } = 56 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 924 }{ 56 } = 33 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 924 }{ 65 } = 28.43 ; ;
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{ 33**2-56**2-65**2 }{ 2 * 56 * 65 } ) = 30° 30'37" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 56**2-33**2-65**2 }{ 2 * 33 * 65 } ) = 59° 29'23" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 65**2-33**2-56**2 }{ 2 * 56 * 33 } ) = 90° ; ;
6. Inradius
T = rs ; ; r = fraction{ T }{ s } = fraction{ 924 }{ 77 } = 12 ; ;
7. Circumradius
R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 33 }{ 2 * sin 30° 30'37" } = 32.5 ; ;