In Exercises 17–32, two sides and an angle (SSA) of a triangle...

Question

In Exercises 17–32, two sides and an angle (SSA) of a triangle are given. Determine whether the given measurements produce one triangle, two triangles, or no triangle at all. Solve each triangle that results. Round to the nearest tenth and the nearest degree for sides and angles, respectively.
a = 10, b = 30, A = 150°

a = 10, b = 30, A = 150°

Accepted Answer

Hey, everyone in this problem, we're asked to identify whether the following measurements of two sides in an angle can produce one triangle, two triangles or no triangle. And to find the missing side length and angles for the triangle. And we're told to express the lengths and angles to one decimal, we're given side length A is equal to 15 side length B is equal to and angle A is equal to 148 degrees. We have four answer choices. Option A and B both have that this can produce one triangle and they have different values for the angle B, angle C and side C which we're gonna come back to as we work through this problem. Option C is that there are two triangles and option D is that there is no triangle. So let's think about that. No, if we're thinking about our sign law, then let's first think about this ambiguous case of the sign law to figure out if we could have to try. Now, angle A which were given us 148 degrees is not acute. This angle is bigger than 90 degrees. So this does not fit the requirement for the ambiguous case of the sign law, which means we have at most one trend. OK. So because A is not a cute, we definitely don't have two triangles. We're gonna cross out option C and we're looking at one of the other options. Yeah. Now let's think about our side here. We have two sides in a corresponding angle. We wanna find the other. Now, one thing to know is that angle A is greater than degrees. We know that and we've already mentioned that this is not an acute angle. So it's greater than 90 degrees. But we also know that the side B is greater than side A. If side B is greater than side A, then angle B has to be greater than angle A. They recall that in a triangle, the opposite side corresponds to that opposite angle. Angle B can't be greater than angle A because we have a maximum of 180 degrees in our time. OK. So this is not possible. So we're gonna expect that we do not have an answer. OK? We're not gonna be able to make a triangle out of these values that we were given. But let's go ahead and do this algebraically. OK? If you didn't notice it, this off the bat, you could go ahead with the sign law and you would find the same answer hey, algebraically. So let's go ahead and do that. And recall that the sine law tells us that the ratio of a side length to the sign of its corresponding angle is equal throughout all three sides of a triangle. So in this case, we're gonna write A divided by S sign of A is equal to B divided by S sign of B. We know both side links and B, we know angle A. So the only unknown here is angle B OK. So that's where we're starting. We're gonna start by calculating that angle B and then we can move on to the other dimensions that we're missing. Substituting in what we know we have 15 divided by sine of 148 degrees is equal to 45 divided by sign that's B rearranging multiply both sides by sine of B. You get that sign of B it's gonna be equal to. Now dividing by 15 multiplying by sine of 148 degrees. We have sin of B is equal to sine of degrees multiplied by divided by 15. Simply simplify the right hand side substituting that into a calculator to get that sign B is approximately equal to 1.59. Now this is a problem we know that sine has a maximum value of one. OK. So we cannot take sign of something and get 1.59. OK. Again, this is not possible. There is no solution for B which tells us that there is no triangle that we can make out of the values we were given. So if we go back to our answer choices, we are going to go with answer choice. D here. No triangle. Thanks everyone for watching. I hope this video helped see you in the next one.

Key Concepts

Law of Sines

Ambiguous Case of SSA Triangles

Triangle Angle Sum Property

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