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 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.

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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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)

Sources:

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