Find the length of each side labeled a. Do not use a calculato...

Question

Find the length of each side labeled a. Do not use a calculator.

Accepted Answer

Welcome back. Everyone in this problem, we want to find the missing side length of the triangle ABC. For our answer choices A says it's 21 centimeters. B 42 centimeters, C 10.5 centimeters and D 30 centimeters. Now, if we're going to find the missing side length of the triangle ABC, it probably would help for us to first think about the value of our missing angle. Angle B. OK. Now, if we think about it based on the angle, some property of triangles, all the angles in a triangle sum to 180 degrees, we already know two of those angles, 120 degrees A and C 30 degrees. So that means the remaining angle B must be the difference between 180 degrees and the sum of 120 30 or 150 180 minus 150 equals 30 degrees. So B is 30 degrees and that makes sense because 120 plus 30 plus 30 equals 180. So we know the value of angle B. Now that we know the value of an B, let's ask ourselves, what else do we know about the triangle? Well, notice that again, angle B is directly in front of side AC which is 21 centimeters. So we know our angle and we know the side, the value of the side opposite to that angle. We're looking for the missing side length A B and for that missing side length, we know the angle opposite to A B. So now let's ask ourselves, do we know anything in a triangle that relates the opposite sides of the triangle to the angle? Well, we can recall the sign law, OK. We call the sign law because that's one way to relate them. Because recall that by the sine law, it tells us that the ratio of the sine of an angle to its opposite side is the same throughout the triangle. In other words, if I had a triangle ABC, then the side A to the sine of angle A is going to be the same as the side B to the sine of angle B, which is the same as the side C to the sine of angle C. OK. So in this case, if we label the opposite sides for our ankles, that is label BC side A label AC side B and label our unknown side A B as side C, OK, then we can use what we know about side B and the sine of angle B to help us figure out our unknown side C because by the sine law, then that means that the value of side B, side C which is A B or the ra of side C, which is A B two, the sine of 30 degrees is going to be the same as the ratio of side B which is 21 centimeters to the sign of the opposite to it, which is 30 degrees. Now, with this formula, we should be able to solve for A B. Now take a good look at both sides of the formula. What do you notice? Well, on the right hand side, the numerator is 21 centimeters. Ok. On both sides, they have a denominator of sine 30. So if their denominators are the same, that must mean their numerators are the same that tells us, then that A B has to be equal to 21 centimeters A is the correct answer. Thanks for watching everyone. I hope this video helped.

Find the length of each side labeled a. Do not use a calculator.
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Key Concepts

Triangle Properties

Trigonometric Ratios

Geometric Relationships

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