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

Question

Accepted Answer

Welcome back. Everyone. In this problem, we want to determine the missing length of the triangle. For our answer choices A says it's two multiplied by the square root of three B three, multiplied by the square root of two C four, multiplied by the square root of three and ad four multiplied by the square root of two. Now, for our triangle, let's just label it to refer to it a bit easier while we, while we solve. So let's label this corner as P and then Q and then R OK, which means then that the side opposite to angle P, 60 degrees will be common P, the side opposite to angle Q will be common Q. So Q equals two and opposite to R is common, RR equals four. And we're trying to figure out the length of side P. Now, what do we know here that can help us? Well, first, let's think about the information that we have. So we have two sides, Q and R OK? And we have the angle included between them angle P which is 60 degrees. So if we have a side, an angle and a side, what can help us to figure out the unknown side. Well, we can think about the law of Core Sciences and recall that the law of co science tells us that if we have two sides and an angle between them, OK, then the square of the side in front of or opposite to the angle is going to be equal to the square of the other two sides. So that's Q squared plus R squared minus two multiplied by lower sides. Q are multiplied by the cosine of the anger between them. In this case, an P know that we have the value of Q and angle P, we can substitute it to solve for our unknown side P. So that means P squared is going to be equal to two squared plus R squared. So that's four squared minus two, multiplied by two, multiplied by four, multiplied by the cosine of 60 degrees. Now, we know that two squared is 44 squared is 16, two, multiplied by two, is four and four, multiplied by four is also 16. And we also know the cosine of 60 is equal to a half. OK. And now if we simplify this further 16, multiplied by a half equals it and four plus 16 minus it is 20 minus eight, which is 12. So P squared equals 12. That means if we want to find P, we can square both sides which tells us that P equals the square root of 12. But we can simplify this a bit further because recall, OK, we also know that the square root of a product is equal to the square or sorry, the product of both of those individually, both of those terms in the product individually. In other words, the square root of A B equals the square root of a multiplied by the square root of B. So in this case, we can rewrite the square root of 12 if we can get a square number out of it and four is a square number. So that means we can rewrite P as the square root of four multiplied by three, which is going to be the square root of four multiplied by the square root of three. And the square root of four is equal to plus or minus two. So plus or minus two multiplied by the square of three. Now, remember we're talking about the sides of a triangle and the length of the side cannot be negative. It's going to be the absolute value. So therefore, that means P is equal to two multiplied by the square root of three. That's the length of the missing side. If we look back on our answer choices, it is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

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

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Verified Solution

Key Concepts

Triangle Properties

Trigonometric Ratios

Law of Sines and Law of Cosines