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.A = 56°, C = 24°, a = 22

Accepted Answer

Hey, everyone in this problem, we're asked to find missing sidelights in angle of the triangle. ABC, we're told that angle A is equal to 46 degrees. Angle C is equal to 34 degrees and side length A is equal to 18. We're asked to express the lengths to one decimal place and the angle to the nearest degree. We're given four answer choices A through D, each of them containing different values for the three missing values that we're looking for. OK. So what we're looking for is that lengthy side, Lake Sea an angle B? Now let's start with angle B. OK. We're already told two angles. We know that the sum of the angles in a triangle adds up to 180 degrees. And so we can use that information to calculate angle B right off the bat. OK. So we can write that angle A what's angle be plus angle C is gonna be equal to 180 degrees. And because this is a triangle substituting in our values, we have 46 degrees course angle be plus 34 degrees is equal to degrees. 46 degrees plus 34 degrees gives us degrees right on the left hand side. If we subtract that from both sides, we get that angle B is equal to 100 degrees. All right. So we found angle B to be 100 degrees. Looking at our answer choices, the only answer choice with angle B being 100 degrees is option C. So we expect option C to be the correct answer. if you were on a test and you had a limited amount of time, you could answer C and move on with this que, move on with the rest of your test. In this case, we're gonna go through the rest of the problem. Wanna make sure that we understand how to find those sidelines. And it's also good to go through and just make sure you made no mistake in calculating angle B OK. There are some answers that are quite close. So we just want to double check that we've made no mistakes. Now we have angle B OK? We have angle A, we have angle C, all the angles and we have a corresponding side to angle A. So what we wanna do is think about our law of sites and our law of signss tells us that the ratio of a side length to the sign of the corresponding angle is consistent or equal for any side and angle in our trunk. Let me write that down in terms of what it means in this problem, OK. It means that side like a divided by a sign of angle A is gonna be equal to side length B divided by sine A or B. Now, I've chosen to relate B to A because we know side length A and we know angle A. OK. So we know the entire left hand side, we know angle B. So the only unknown we have is side length B OK? So that's why we related it to A because we knew that entire ratio substituting in our values, we have that 18 divided by sine of 46 degrees is equal to B divided by a sign of 100 degrees. We're gonna multiply both sides by sine of 100 degrees to isolate for B B is equal to 18 multiplied by sine of 100 degrees divided by sign of 46 degrees. And if we work this out on our calculator, we get that this is approximately equal to 24.643 units. Rounding to one decimal place. This is gonna be about 24.6 units and that matches the answer choice C value for B as well. So that's looking good. OK. That answer we're expecting to be correct so far matches and now we need to move on to calculating side C and we're gonna use A again. OK? You could relate it to B now because we've just found the side length of B but this value is rounded a little bit, there's a little bit of round off error there. So we want to avoid that if we can. So we have that a divided by sine of angle A is equal to C divided by a sign of Anglesey. OK. Again, we get 18 divided by sine of degrees. It is equal to C divided by a sine of degrees multiplying both sides by sine of 34 degrees. We get that C is equal to 18 multiplied by a sine of 34 degrees divided by sine of 46 degrees. In approximating this using our calculator. We get that C is approximately equal to 13. units. If we round that to one decimal place, that's gonna be 14.0 units which again matches with option C. So we found angle B to be 100 degrees. Side B to be 24.6 and side C to be 14.0 units. And so the correct answer here is option C. 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.A = 56°, C = 24°, a = 22

Key Concepts

Law of Sines

Triangle Sum Theorem

Rounding Rules

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