Posted by ken on Wednesday, June 9, 2010 at 8:55am.
find angle C first of all
sinC/50 = sin53/40
sinC=.99829
angle C = 86.65° or 93.35° (we have the ambiguous case, you can actually draw two triangles satisfying your information.
then angle B = 40.35° or 33.65°
Case 1
AB=50
BC=40
angle B = 40.35
area = (1/2)(50)(40)sin40.35
= 647.41
Do the second case in the same way, you should get 554.16
According to the law of sines,
40/sin 53 = 50/sin C
Therefore
sin C = (50/40)*sin53 = 0.998
C = 86.7 degrees
B = 180 - 53 - 86.7 = 40.3 degrees
With all sides and all angles, you can solve for the area. If you pick AC as the base,
Area = (AC)*(1/2)*(BC)*sin 86.7
= 50*(0.5)*40*.998 = 998 m^2
Since my numbers don't agree with Reiny's, check mine for mistakes. Reiny is right about there being two cases. I assumed all acute angles
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