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.A = 162°, b = 11.2, c = 48.2

Accepted Answer

Hey, everyone in this problem, we're asked to find the solution for the following triangle given the measurement of one angle and two sides. If there is no possible triangle, we're asked just to indicate no triangle. And if two triangles exist, we have to solve each one. We're asked to round the resulting angle and side measurements to one decimal place. So what we're given is that angle B is equal to 100 and 43 degrees side A is equal to 13.9 and side C is equal to 55.6. We have four answer choices A through D in this problem. And we're gonna come back to those as we work through this problem. So let's start with the first question. And the first question is, do we have two triangles, one triangle or no triangle? Now, in this case, we're given two sides and the enclosed angle. OK. We don't have a side and the opposite angle. So what that means is that the ambiguous case does not apply. OK. So we have a maximum one triangle. If we look at our answer choices, option A has that we have two different triangles, we know that that's not possible. So we can already eliminate option A again because that ambiguous case does not apply. So now we're dealing with at least or sorry, at most one triangle, let's try to draw it out and see what we can do. So we can draw this triangle. We know that we have angle B being 143 degrees across from angle B, we will have side B on the left, we will have angle A across from angle A, we have side A which we know is 13.9. And then we have angle C and across from angle C, we have side C 55.6. OK. So this is what our triangle looks like or we think it looks like we need to figure out if there is an A and C valued for angles and side B that make this triangle true. So again, we have two side links in an enclosed angle. We can use that along with the law of cosines to find side B. So let's start by doing that. OK. And we're called it, the law of cosines tells us that the side we're looking for. So in this case, B squared, it's gonna be equal to the sum of the other side squared A squared plus C squared minus two, multiplied by the other two sides. So multiplied by A multiplied by C multiplied by cosine of the opposite angle B we can substitute this in with the values for our problem. So we get B squared is equal to 13.9 squared plus 55.6 squared minus two, multiplied by 13.9 multiplied by 55.6 multiplied by cosine of 143 degrees. If we simplify everything on the right hand side, we get that B squared will be equal to 4519. And we can take the square root to find that B is gonna be equal to 67.2235 approximately. So we have side length beef. Now we need to do angles A and angle C. Let's go ahead and start with angle A and how can we do that? Well, we know side A and now we know side and angle B that allows us to use the law of sign. So recall that the law of signs tells us that sign of angle A divided by side A is equal to sine of angle B divided by sine side B filling in what we know we get sine of angle A divided by 13.9 is equal to sine of 143 degrees divided by side B which we just found to be 67.2235. We can multiply both sides by 13.9. So we get that sign of angle A is gonna be equal to 13.9 multiplied by s of 143 degrees divided by 67.2235. We can work out the right hand side and then take the inverse sign. And what we're gonna find is that angle A will be equal to about 7.148 degrees. OK. So we have angle A. Now we already found side B last thing we need to do is find angle C in angle C we can do in the exact same way we did angle A, we're gonna use a law of signs. But now instead of comparing A and B, we're gonna compare B and C. So we can say that sign of angle C divided by side C will be equal to sine of angle B divided by B. OK. So let's take a look at our diagram for a minute just to make sure we understand. OK. So we're looking for Anglesey, we know the opposite side and we know both angle B and psyche. OK. So we have everything we need to solve for this missing angle substituting in what we know. We get. That sign of C divided by 55.6 is equal to sign of 143 degrees divided by 67.2235. We can multiply both sides by 55.6. We get that sign of C is equal to sign of 143 degrees. Multiplied by 55.6 divided by 67.2235. Again, working at the right hand side and taking the inverse sign, we get that angle C is going to be equal to 29 0.85 degrees. So that's it. We now have the missing angles and the missing side. And we found that this works out. So we do get a triangle out of the information we were given. So we can eliminate option D that we have no triangle. We're left with option B or C. And if we compare with the values, we found we'll see that the correct answer is going to be option B. So side B is 67.2 units, angle A is about 7.1 degrees and angle B is about 29.9 degrees. 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.A = 162°, b = 11.2, c = 48.2

Key Concepts

Law of Sines

Triangle Existence Criteria

Ambiguous Case of SSA

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