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 = 30, b = 20, A = 50°

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 also find the missing side lines and angles of the triangle expressing to one decimal place. So we're given side A is 35 side B is 28 and angle A is 70 degrees. We have four answer choices A through D and we're gonna come back to these as we work through the problem. So let's start with this question of how many triangles can we have? OK. Taking a look at the information we were given, we can see that side A is greater than side B. And what that means is that the ambiguous case does not apply. And if the ambiguous case does not apply, then we get at most one track. OK. So taking a look at our answer choices. Option A and B have that we get one triangle. Option C has that we get two triangles. So we can already eliminate option C. We know that we have at most one triangle. So we can't have two option D has that we have no triangle. That's still a possibility. And we have at most one triangle, but it is possible that we won't be able to make a triangle at all. So let's keep going with this so that we can narrow down those answer choices even further. So we have a side in the corresponding angle. So we can use that along with the sine law in order to find another one of these missing angles. So we call that sign law tells us that sign of angle A divided by sine side A sorry is going to be equal to sine of angle B divided by C. We have side A and B, we have angle A. So the o only unknown in this equation is angle B, we can substitute in our values. And so when we do that, we get sign of 70 degrees divided by 35 is equal to sign of angle B divided by 28 rearranging sine of angle B is going to be equal to 28 multiplied by sine of 70 degrees divided by 35. And finally, we can take the inverse sign on both sides and we'll get that angle B is approximately equal to 48.74 degrees. So we have, and we have two more things to find. We have side Sea and we have Anglesey. So what we're gonna do is we are actually going to start with Anglesey because it's a little bit simple and it's simpler because we know that the angles in a triangle have to add up to 180 degrees. So angle A plus angle B plus angle C must be equal to 180 degrees. We know that angle A is 70. Angle B is 7 48.74. Sorry. And so we have that 70 degrees plus 48.74 degrees plus angle C must be equal to 180 degrees. We can subtract 70 48.74 from both sides. And we get that angle C is going to be equal to about 61.26 degrees. OK. So we're done with our angles. We have all of the angles in our triangle. We have one side left to find. Now, just like we did when we were finding angle B, we're gonna use sign log this time, we'll know both angles and we'll be looking for a side instead, but it works very much the same way. So let's now relate A to C and so sign of angle A divided by side A is equal to the sine of angle C divided by sighs. Now, we could technically use B A sign of B divided by B and relate that to C. We're using A because that's the information we were given in the problem. So we can avoid a little bit of round off error. OK. In the angle we found for B there's a little bit of round off error in the calculations that we did. Um So just using the values we were given if we can, helps us avoid that. OK. So substituting in our values sign of 70 degrees divided by 35 is equal to sign of 61.26 degrees divided by sightsee. So side C is going to be equal to 35 line of 61.26 degrees divided by sign of 70 degrees. And when we work this out, we get that side C is going to be equal to 32.658 approximately. OK. So we have all of the values we needed. We found the missing angles. We found the missing side. We can now go back and answer this question. Now, option D, we can eliminate option D was that there was no triangle, we could make, we just found one that we can, we found the angles, we found the side links, it all works out. So there is a triangle so we can eliminate. Option D, it comes down to A and B. The only difference is these values. So option A, we have that angle B is 48.7 degrees. That's what we found. Angle C 61.3 degrees. That's again what we found and side C is 32.7 units which again, when we round is what we found and so the correct answer here is going to be option A. Thanks everyone for watching. I hope this video up, see you in the next one.

Key Concepts

Law of Sines

Ambiguous Case of SSA

Triangle Sum Theorem

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