In Exercises 1–12, solve each triangle. Round lengths to the near...

Question

In Exercises 1–12, solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. If no triangle exists, state 'no triangle.' If two triangles exist, solve each triangle.

B = 107°, C = 30°, c = 126

Accepted Answer

Hey, everyone in this problem, we're asked to identify whether the measurements of two angles in one side that are given can produce one triangle, two triangles or no trend. We were asked to find the missing side length and angles of the triangle and to express those lengths to one decimal place and the angles to the nearest degree, we're told that angle B is equal to 101 degrees. Angle C is equal to 45 degrees and side C is equal to 132. Option A and B both have that the answer is one triangle and they just differ in the values for the missing side links and angle. Option C is that we can make two triangles. And option D is that we cannot make a triangle with the information we were given. Now looking at the information we were given, OK, we were given two angles B and C. Now recall that the sum of the angles in a triangle add up to 180 degrees. So if we know two of them, we can find the missing angle. And so we know the angle A what A will be plus angle C is going to be equal to 180 degrees. So angle A plus 101 degrees plus 45 degrees is gonna be equal to 180 degrees. Subtracting 101 and 45 degrees from the left hand side A to move them to the right, subtracting them from the right. Forget that A is equal to 34 degrees. OK? So right off the bat, we were able to find our angle A. So it's important to note here is because we were given two angles of our triangle that automatically defines that third one. And so we can only have one triangle and we cannot have two different triangles where we have the same angles. OK? So we know that we can eliminate option C it's not possible to make two different triangles when we were already given two angles. All right. Now what's our next step? We want to find these missing sides A and B and we know side C and angle C. OK. And then we know both of the other angles. So let's use our law of signs, OK. We're called that our law of signs allows us to relate the ratio of a side to the sign of its corresponding angle in a triangle. OK? And it tells us that that ratio is gonna be the same for any side, any angle that we choose. OK. So in other words, we can write OK, divided by a sign of angle A is equal to side B divided by sign of angle B, which is equal to side C divided by a sign of Anglesey. And when we're looking at this expression, let's think about the values. We know, we know side C and we know the angle C because we're gonna get green check marks on both of those values. And then we also know angle B and angle A. OK. So when we're populating side A and side B, we're gonna want to compare them to C because we know all of the information about B. So let's go ahead and do that. And we're gonna do each side one at a time. So starting with A, we have a divided by sign A is equal to C divided by sin of C substituting inner values A divided by sign of 34 degrees is going to be equal to 132 divided by a sign of 45 degrees. If we multiply both sides by sine of 34 degrees, A is gonna be equal to 132 fine of 34 degrees divided by sine of 45 degrees, which gives us site A of approximately 104.388. OK. So we're done with side A. Now we move to side B. Now for some B, we could technically compare it to either A or C OK? Because we have all of the other information now, but we're gonna compare it to c because that's the information we were given and there's two reasons for that. Ok. The first is to avoid any round off error. And when we wrote down the answer for side a 104.388 we rounded off that decimal place a little bit. Ok. So there's a tiny bit of round off error there. We don't want to carry that forward. And the second reason is just in case we made a mistake in our calculation for a, if we use the information about C that we were already given, we don't have to worry about carrying that error forward either. OK. So we're gonna compare B to C side B divided by sign of angle B is equal to side C divided by sine of angle C substituting in the values we have B divided by sine have 101 degrees is equal to 132 divided by a sign of 45 degrees in multiplying both sides by sine of 101 degrees. We have 132 multiplied by sine of 101 degrees divided by sine of 45 degrees. Which one we work this out on a calculator gives us B equal to 183. approximately. Now we can double check that our answers make sense. And we know that the smallest side will always be across from the smallest angle, same thing from the largest side and the largest angle. OK. So the smallest angle we had was a 34 degrees. And we found a to be the smallest side, it's um angle B, it was the biggest angle we had. And we found side B to be the biggest side. OK. So that makes sense in terms of the relationship between the three. And so we go back to compare to our answer choices and we round like the question has asked us to, we found that we have one triangle side A was 104.4 units. Side B was 183.2 units and angle A was 34 degrees. This corresponds with answer choice. A thanks everyone for watching. I hope this video helped see you in the next one.

In Exercises 1–12, solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. If no triangle exists, state 'no triangle.' If two triangles exist, solve each triangle. B = 107°, C = 30°, c = 126

Key Concepts

Triangle Sum Theorem

Law of Sines

Ambiguous Case of the Law of Sines

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