How do you solve the triangle ABC given a=55, b=25, c=72?

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Answer 1 · 2016-08-24T17:09:34

#hatA = 39.3°, hatB = 16.8° , hatC = 123.9°#

"Solve a triangle" means find all the missing information - in this case, the size of each angle.

The lengths of 3 sides are given. We must use the Cosine Rule.

Find the biggest angle first - if it is an obtuse angle, the Cos rule will indicate this, but the Sin Rule will not,

Angle C is the biggest angle because it is opposite the longest side.

#Cos hatC = (a^2 + b^2 - c^2)/(2ab) = (55^2 +25^2 -72^2)/(2xx55xx25)#

#Cos hatC = -0.5578" (the negative value means it is obtuse)"#

#color(red)(hatC =123.9°)#

Now use the SIn Rule.

#(Sin hatA)/a = (Sin hatC)/c#

#Sin hatA = (55 xx Sin 123.9)/72 = 0.6340#

#color(red)(hatA = 39.3°)#

We have 2 angles, use the angle sum of a triangle to find #hatB#

#hatB = 180° - 123.9° - 39.3°#

#color(red)(hatB = 16.8°)#