The ratio of the angles of scalene triangle is alpha:beta:gamma=1:4:7 (angle), the lenght of side c is 20cm. Determine the size of the angles and the length of the unknown sides. Find the are of the triangle.

The problem provides the information that the ratio of the angles of triangle is `alpha:beta:gamma = 1:4:7` , such that:

`alpha/beta = 1/4 => beta = 4 alpha => alpha = beta/4`

`beta/gamma = 4/7 => 4gamma = 7beta => gamma = (7beta)/4`

You need to remember that the sum of angles of triangle is of `180^o` , such that:

`alpha + beta + gamma = 180^o`

Replacing `beta/4` for alpha and `(7beta)/4` for gamma, yields:

`beta/4 + beta + (7beta)/4 = 180^o`

`(8beta)/4 + beta = 180^o => 2beta + beta = 180^o => 3beta = 180^o => beta = 60^o`

`alpha = 60^o/4 => alpha = 15^o`

`gamma = 7*15^o = 105^o`

You need to use the law of sines to evaluate the unknown lengths, such that:

`c/(sin gamma) = b/(sin beta) = a/(sin alpha)`

`20/(sin 105^o) = b/(sin 60^o) = a/(sin 15^o)`

`20/(sin(90^o + 15^o)) = b/(sqrt3/2) = a/(sin 15^o)`

`20/(cos 15^o) = 2b/sqrt3 = a/(sin 15^o)`

`20/(cos 15^o) = 2b/sqrt3 => 10/(cos 15^o) = b/sqrt3`

`cos 15^o = cos(45^o - 30^o) = (sqrt6 + sqrt2)/4`

`sin 15^o = sin (45^o - 30^o) = (sqrt6 - sqrt2)/4`

`20/(cos 15^o) = a/(sin 15^o) => a = 20 tan 15^o`

`a = 20* (sqrt6 - sqrt2)/(sqrt6 + sqrt2)`

`a = 20*(sqrt6 - sqrt2)^2/(6 - 2)`

`a = 5(sqrt6 - sqrt2)^2 => a = 5(8 - 4sqrt3) => a~~5.358 cm`

`10/(cos 15^o) = b/sqrt3 => b = 40sqrt3/(sqrt6 + sqrt2)`

`b = 10sqrt3(sqrt6 - sqrt2) => b = 30sqrt2 - 10sqrt6=> b~~17.925 cm`

Hence, evaluating the unknow angles and unknown lengths of the given scalene triangle, using the law of sines, yields `alpha = 15^o, beta = 60^o, gamma = 105^o, a~~5.358 cm, `` b~~17.925 cm.`

(sqrt6 + sqrt2)