Solve each triangle ABC.

Question

Accepted Answer

Welcome back. Everyone in this problem, we want to find the missing side length and angle of the triangle. ABC. For our answer choices, A says angle A is 27.5 degrees. Side BC is 165.4 centimeters and side AC is 101.6 centimeters. B says A angle A is 31.5 degrees BC is 102.9 centimeters and AC is 156.4 centimeters. C says angle A is 27.5 degrees. BC is 101.6 centimeters and AC is 165.44 centimeters and D says angle A is 27.5 degrees BC is 98.2 centimeters and AC is 137.8 centimeters. Now, if we look at the triangle, we have here, first, let's identify our missing side L OK. So we know our missing angle is A and for the sake of simplicity, when referring to a triangle, let's label our sides the typical way we usually label triangles. That is if we have an angle A then the side opposite to anger A is common, a side opposite to angle B, we label as common B and the side opposite to angle C, we label as common C. So in this case, cr A is 79.6 centimeters. Now notice that we have two angles in our triangle so we can find the third one before we go any further. So let's solve for angle A which is really the first part of our problem. OK. So solving for a, what do we know? Well, we know that angles in a triangle sum to 180 degrees. So since angles in a triangle, since we know that fact, we can use that idea to solve for angle A because to find angle A, all we need to do now is to subtract angles B and C from 180 degrees. So that's going to be 180 degrees minus 131.2 degrees and 21.2 degrees. OK? No, 131.2 plus 21.2 is going to be 152.4 and 180 minus 152. Oops, this should be 131.3 ami bad guys. It's 131.3. So let's let's rewind, let's do that again. So that's 131.3 plus 21.2 which is gonna be 152.5 degrees and 180 minus 152.5 is going to be equal to 27.5 degrees. So an A is 27.5 degrees. Let's write that on our triangle. Now we know the value of angle A next, we need to figure out our side BC and AC notice here, OK. That we know we know angle C 21.2 degrees and we know the side opposite to angle C 79.6 centimeters. OK. So we know a pair of angle and opposite side. What do we know about triangles that relates a pair of a known pair of opposite side and an angle? Well, we can recall the sign law. OK? Because the sign law tells us, OK, that the ratio of the sine of an angle to its opposite side is the same throughout the triangle. In other words, the ratio of S A to the sine of A is the same as the ratio of side B to the sine of B, which is the same as the ratio of side C to the sine of C. So if we know a pair of side and angle, then we should be able to figure out any other pairs once we know at least either the angle or the value of the side. And in this case, we do know a pair of angle and side, as we said, we know side C is 79.6 centimeters and angle C is 21.2 degrees. So using that idea, then we should be able to figure out the value of BC which is A and the value of AC, which is B because now by applying the sine rule, that means then that A which is side BC two, the sine of 27.5 degrees. Yeah, is going to be equal to side C which is 79.6 centimeters to the sign of the anger opposite to it, the sign of 21.2 degrees. Now we can solve for BC if we multiply both sides by the sine of 27.5 degrees. When we do that, we get BC to be equal to 79.6 multiplied by the sine of 27.5 degrees. OK? All divided by the sine of 21.2 degrees. OK? So now we can use that to figure out our side BC. We can put this expression into our calculator. Now, when you do that, OK, when you do that, you should get BC to the nearest 10th of A centimeters like it's written in our answer to be equal to 101.6 centimeters. OK? So that's the value of BC. Likewise here, if we try to solve for side AC, remember that side AC is also side B and using the sine ratio again, that means AC, side B to the sine of angle B which we know is 131.3 degrees is going to be equal to sc we can, I mean, we could use A and sine A but let's just use C and sine C since we have it here. So it's gonna be equal to 79.6 centimeters divided by the sine of 21.2 degrees. Now, if we, again, if we multiply both sides by the sine of 131.3 then AC is going to be equal to 79.6 multiplied by the sine of 131.3 degrees divided by the sine of 21.2 degrees. That's the value of AC, if we put that expression into our calculator, OK, then we should find that AC to the nearest 10th of a centimeter is equal to 165.4 centimeters. Therefore said, sorry A is 27.5 degrees. BC is 101.6 centimeters and AC is 165.4 centimeters. If we look back on our answer choices, that means C is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Solve each triangle ABC.

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Key Concepts

Triangle Classification and Properties

Law of Sines and Law of Cosines

Trigonometric Ratios and Angle Measures

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