How do you determine the number of possible triangles and find the measure of the three angles given #a=9, c=10, mangleC=150#?

1 Answer
Oct 31, 2017

#A=26^@45', B=150^@, C=3^@15'#

Explanation:

Since the given information is for a SSA triangle it is the ambiguous case. In the ambiguous case we first find the height by using the formula #h=bsin A#.

Note that A is the given angle and its side is always a so the other side will be b .

So if #A < 90^@# and if

  1. #h < a < b# then then there are two solutions or two triangles.

  2. #h < b < a# then there is one solution or one triangle.

  3. #a < h < b# then there is no solution or no triangle.

If #A >=90^@# and if

  1. #a > b# then there is one solution or one triangle.

  2. #a <=b# there is no solution

#h=9 sin150^@=4.5#, since #4.5 < 9 < 10 # we have

#h < b < a# so we are looking for one solution. Hence,

#Sin A/a = sin B / b#

#sin A /9 = sin 150^@/10#

#sin A =(9 sin 150^@)/10#

#A=sin^-1 ((9 sin 150^@)/10)=26^@45'#

and therefore

#C=180^@-150^@-26^@ 45'=3^@15'#