In triangle ABC, A = 30°, C = 105°, a = 8. Find side b.

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

In a triangle ABC, a/sin A = b/sin B = c/sin C

The angles of a triangle add up to 180 degrees.

A = 30 degrees and C = 105 degrees, this gives B = 180 - 30 - 105 = 45 degrees.

a/sin A = b/sin B

=> 8/sin 30 = b/sin 45

=> b = 8*sin 45/sin 30

=> b = 8*2/sqrt 2

=> b = 8* sqrt 2

=> b = 11.31

The side b is 11.31

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

To determine the length of the side b, we'll apply the law of sines:
a/sin A = b/sin B
We notice that we do not know the measure of the angle B.
Since we know two of the three angles of the triangle, we can determine the 3rd angle, knowing that the sum of the angles is of 180 degrees.
A+B+C = 180
30+B+105 = 180 => B = 180 - 135
B = 45
Now, we can apply the law of sines:
8/sin 30 = b/sin 45
We'll cross multiply:
b*sin 30 = 8*sin 45
b/2 = 8*sqrt2/2
b = 8sqrt2


The requested length of the side b is: b = 8sqrt2 units.

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