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 = 57.5, c = 49.8, A = 136°

Accepted Answer

Hey, everyone in the scroll were asked to identify whether the following measurements of two sides in an angle can produce one triangle, two triangles or no triangle. And then we're asked to find the missing side links and angles for this triangle. We're told to express those to one decimal place. We're told side length A is 45.3 side length C is 37.6 and angle A is 128 degrees. We're given four answer choices, options A through D each containing different combination of answers for the different parts of this question. And we're gonna come back to these as we work through the problem. So the first thing we're gonna do is try to figure out this question of how many triangles we have. And what we're gonna do is look at angle A, an angle A that we're given is 128 degrees. This is not an acute angle. OK? It's greater than 90 degrees. And so what that means is the ambiguous case does not apply. And if the ambiguous case does not apply, we have at most one, right. So if we take a look at our answer. Choices. Option A and B have that we have one triangle. Those are possibilities. Option C that has that we have two triangles and we know that's not possible. We have at most one triangle so we cannot have two. So we can eliminate option C already. Option D is that we have no triangle. Again, this is a possibility. We know we have at most one triangle, but it is possible that we can't form a triangle with what we were given. We'll leave that answer for now and we'll do some more work to figure out what the correct answer is gonna be. So now let's move on to finding these side links and angles were given corresponding side length A and angle A, we have another side link C. So we can use the sign law to determine angle C. So we recall that sin law tells us that the ratio of sine of an angle to its corresponding side is equal for all three sides in our triangle, all three angles in our triangle. So we can write that sign of A divided by side A is equal to sign of angle C divided by side seat. We wanna solve for angle C. So we can rearrange, we're gonna have that sign of angle C is equal to C multiplied by sine of angle A divided by A. We have all the values we need on the right hand side. Let's go ahead and substitute them in and we're gonna have that sign of Anglesey is equal to 37.6 multiplied by a sign of 128 degrees divided by 45.3. You can take the inverse sign on both sides. And what we'll find is that angle C is going to be equal to approximately 40.849 degrees. Hi. So we have angle C, we have side C, we have both angle A and side A. So now we're left with figuring out the information for the other angle and inside we'll call it B hm. What we're gonna do first is figure out missing angle B and we're gonna do the angle because it's gonna be the simplest calculation. They recall that the three angles in a triangle add up to 180 degrees. So we must have that angle A plus angle B plus angle C is equal to 180 degrees. We know that angle A is 128 degrees. We know angle C is 40.849 degrees. So we have 128 degrees lost angle B plus 40.849 degrees is equal to 180 degrees. We can subtract 128 degrees and 40.849 degrees from both sides. And we're gonna get that angle B is approximately equal to 11.151 degrees. So we have angle C, we have angle B. The last thing we need to find is side length B and just like we did when we found angle C, we can use the sign. OK. So in this case, we're gonna put the side blanks in the numerator instead of the denominators, it's the exact same rule, it works in the exact same way. And we're gonna compare side B and side A. So we have side blank A divided by sign of angle A is equal to side length B divided by sign of angle B. And again, rearranging for the value we want, which is slide B, we get that B is going to be equal to a multiplied by sign of angle B divided by sign of angle. Last thing we need to do is substitute in our values and solve. So we get side length B is going to be equal to 45.3 multiply by sign of 11.151 degrees, all divided by sign of 128 degrees. And when we work this out, we get that side length fee, it is gonna be approximately equal to 11.1. So now we can see that we can make a triangle out of the values we were given. OK. So we can eliminate option D that there was no triangle and we're left with option A or B and it comes down to these values that we found. So we found that we have a one triangle angle B is approximately 11.2 degrees angle C is approximately 40.8 degrees and side B is approximately 11.1 units. And so we have option A is the correct answer. Thanks everyone for watching. I hope this video see you in the next one.

Key Concepts

Law of Sines

Ambiguous Case of SSA

Triangle Sum Theorem

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