Question

How do you solve the triangle given a=8, b=24, c=18?

Answer 1

Three angles are `color(blue)(hat A = 14.63^@, hat B = 130.75^@, hat C = 34.62^@`

Explanation:

Known three sides. To find the three angles.

`A_t = sqrt (s (s-a) (s-b) (s - c))` where s is the semi perimeter of the triangle.

`s = (8 + 24 + 18)/2 = 25`

`A_t = sqrt(25 * (25-8) (25-24) (25-18))= 54.54`

But `A_t = (1/2) b c sin A`

`:. sin A = (2 * A_t) / (b c) = (2 * 54.54) / (24 * 18) = 0.2525`

`hat A = sin ^-1 0.2525 = 14.63^@`

Similarly,
`hat B = sin ^-1 ((2 * 54.54) / (8*18) )= (180 - 49.24) = 130.75^@` since it is an obtuse angle.

`hat C = sin ^-1 ((2 * 54.54) / (8*24)) = 34.62`