In Exercises 9–16, solve each triangle. Round lengths to the near...

Question

In Exercises 9–16, solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree.B = 5°, C = 125°, b = 200

Accepted Answer

Hey, everyone. Welcome back in this problem. We're asked to find the missing side lengths in angle of the triangle. ABC. We're told to express the length to one decimal place and the angle to the nearest degree we're given that angle B is equal to nine degrees. Angle C is equal to 133 degrees and side B is equal to 180. We have four answer choices A through D, all of them just containing different values for the three things. We need to find side A side C and angle A. So we're gonna start with the angle. OK. We know that the angles in a triangle all add up to 180 degrees, we're already given two angles. So right away, we're gonna be able to calculate that third angle. OK. So what we're looking for here is angle A and like I said, the sum of the angles angle A, what's angle B plus angle C is gonna be equal to 180 degrees. This is a triangle. So we have that angle A, it was nine degrees plus 133 degrees is equal to 180 degrees. If we simplify on the left hand side, adding these two together, and then we subtract from both sides to isolate. We get that angle A is equal to 38 degrees. OK. So we found one of our unknowns, we found the angle. If we compare this to our answer choices. A option A has the angle A being 34 degrees. That's not correct. So we can cross that one out. Option B has the angle being 36 degrees also not correct. So we can cross that one out. Option C and D both have the angle being 38 degrees, which is what we found. So we know it's gonna be one of those two, let's work out these side lengths and then we can come back and figure out which one is correct. All right. So we found angle A and remember the other two things we're looking for are side A and side B. OK. Those are our three unknowns that we are looking for. Whoops, not side B, side C, we know. All right. So let's start with side A. We don't know side A. We wanna find it, we know the angle A and we know the angle and side B. So what we're gonna do is use our law of Science. OK. And recall that our law of science when we're talking about A and B is gonna be the following a side A divided by sign of angle A is gonna be equal to side B divided by sign of the angle. OK. So the ratio of a side length to the sign of its corresponding angle is equal in a triangle, OK? For any side length or angle that you choose. Now again, we're trying to find little A, we know the two angles, we know little B so we just need to plug in our values and solve. So we get that A divided by sign of 38 degrees is equal to 180 divided by sign of nine degrees. Now we're gonna multiply both sides by sine of 38 degrees to isolate for little A I gonna get that little A is equal to 180 multiplied by sine of 38 degrees divided by sine of nine degrees, which is gonna be equal to approximately 708 0.4057. So that's our side length A. Now we're gonna do the exact same thing for C. OK? We can use this law of sign again and we can relate the values of C with the values of B. Now, at this point, you could relate C with A or B because we do have the values for A but it's always best to relate the ones that you were told in the problem. OK? If we relate it to B, we don't have to deal with any round off error. And then if we've made a small error in our calculation here, that's not going to affect our second calculation. OK. All right. So side B divided by sign of the angle B is gonna be equal to sight, see divided by a sign of the Anglesey substituting in our values. 180 divided by sine of nine degrees is equal to C divided by sine of 133 degrees. We multiply both sides by sine of 133 degrees to isolate. And we have that C is equal to 180 multiplied by sine of 133 degrees divided by sine of nine degrees. And if we work this out on our calculator, we get that C is approximately equal to 841.526 units. OK? So now we're done, we have angle A, we have side A and we have side C all of the unknowns we were looking for, OK. We already narrowed it down to option C and D based off of angle A. And now if we look at the side links, we found rounded to one decimal place, we can see that the correct answer here is option D. Thanks everyone for watching. I hope this video helped see you in the next one.

In Exercises 9–16, solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree.B = 5°, C = 125°, b = 200

Key Concepts

Triangle Sum Theorem

Law of Sines

Rounding Rules

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