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

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 = 6.1, b = 4, A = 162°

Accepted Answer

Hey, everyone in this problem, we're asked to identify whether the measurements side A is equal to 5.3. Side B is equal to three, an angle A is equal to 155 degrees can produce one triangle, two triangles or no triangle. And also to find the missing sideline and angles of the triangle, we're told to express the lengths and angles to one decimal. We have four answer choices here. Option A and B both have that this produces one triangle and they just differ in what those values for the missing sides and angles are. Option C is that there are two triangles and option D is that there is no triangle that we can make out of these values. Now, let's start with this two triangle thing. Hey, how can we have two triangles? We have to think about our ambiguous case for the signed law. And what you'll notice is that the angle A, we were given A is not acute, it is not less than 90 degrees, which tells us that this does not fit the requirement for that ambiguous case of the sign law, which means we're not gonna have to drink and we're gonna have at most one triangle. And so we can eliminate option C before we really even start. All right, now we're left with these other options. We are given a side link and a corresponding angle A and then another side. OK. So let's use our sign law to find angle B. And recall that our signed law relates the sign of the angle to its corresponding suck. We have side of A divided by A is equal to sin of B divided by B. And that's what we're gonna use. In this case, we know angle A side A and side B, the only unknown is gonna be angle substituting in our values, we have that site of 155 degrees divided by 5.3 is equal to sign B divided by three, multiplying both sides by three and then just rewriting so that our sign of B is on the left-hand side, we get that side speed is equal to three fine of 155 degrees divided by 5.3. If we take the inverse side and simplifying the right hand side with a calculator, we get that B is going to be equal to about 13.84 degrees. All right. So we have this other angle beat. Now let's move on to angle C. Now, for angle C, we don't have any information about the side C. OK. So we can't use the sign law because we don't know the side or the angle. OK. What we can do though is use the information we have about the angles. We know that the angles in a triangle add up to 180 degrees. So we have A plus B plus C is equal to 180 degrees. We know angle A and B, we can substitute those in and solve for C, We get 155 degrees plus 13.84 degrees plus angle C is equal to 180 degrees. Subtracting 155 and 13.84 degrees from the left hand side, moving it to the right hand side, we have the angle C is equal to 11.16 degrees. All right. So we have our two angles. Now we need to move on to this side. OK. It's looking like this is gonna create one triangle, but we're gonna check and make sure that we can find a side length that works for this triangle before we answer this question now that we have angle C, we can use our sine law again to find side C. OK. So for our sine law, this time, we're gonna make the side links at the top instead of the angles just makes it a little bit easier when we're rearranging, but they are both equivalent expressions. So in this case, we have side A divided by sine of angle A is equal to side C divided by sign of Anglesey substituting in our values 5.3 divided by sine of 155 degrees is able to sightsee divided by sine of 11.16 degrees, multiplying both sides by sine of 11.16 degrees. To get that side length C it's gonna be equal to 5. fine of 11.16 degrees invited by sign over 155 degrees. And if we simplify that, we get that C is approximately equal to 2.4273. So now we have our missing angles and our missing side, we know that this creates a triangle. So we're looking at option A or B and we just need to compare these values we found. So we found that angle B was about 13.8 degrees. Angle C is about 11.2 degrees and the side length of C is about 2.4 units which corresponds with answer choice. A thanks everyone for watching. I hope this video helped see you in the next one.

Key Concepts

Law of Sines

Ambiguous Case of SSA

Triangle Sum Theorem

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