In triangle ABC, the given sides are b = 23 and c = 15 and the angle given is A = 27 degrees. I used the cosine law to determine the length of the side opposite the angle and got 11.798... On the answer key that I have it says that there's only one triangle possible but the opposite side is shorter than the adjacent. Shouldn't this be two triangles?
How many triangles in this oblique triangle?
- Thread starter hatetrig
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I can see why you'd think that, but if you draw out the diagram you can see why there's only one triangle possible.
The diagram's not drawn perfectly, but both the red line BC (or a for the side opposite A) and the blue dashed line measure the same (in this case, the missing third case you calculated with the law of cosines). So, there could potentially be two triangles. However, you were given the side AC (or b for the side opposite B) as 23. If you use the triangle made by using the blue dashed line instead of the red line, then point C would have to move, and the length of AC wouldn't be 23 units. Because it violates the given conditions of the problem, this second triangle is not a valid solution.
The diagram's not drawn perfectly, but both the red line BC (or a for the side opposite A) and the blue dashed line measure the same (in this case, the missing third case you calculated with the law of cosines). So, there could potentially be two triangles. However, you were given the side AC (or b for the side opposite B) as 23. If you use the triangle made by using the blue dashed line instead of the red line, then point C would have to move, and the length of AC wouldn't be 23 units. Because it violates the given conditions of the problem, this second triangle is not a valid solution.
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If two sides of a triangle and the angle contained between them is given - it is a unique triangle by S-A-S theorem.
If two triangles fulfill the same conditions - those are congruent!
Thanks for the help! I had a hard time figuring it out because I was so focused on whether or not this side was longer than the other or if bsinA was greater than 'a' that I did not think about the properties of the triangle at all. The diagram helped a lot too.