Solve each triangle. Approximate values to the nearest tenth.

Question

Accepted Answer

Everyone in this problem, we're asked to find the missing measurements of the triangle. We're given triangle RPQ. A side R has a length of seven. Side Q has a length of seven and side P has a length of 11. We are missing all of the angles in this triangle. So that's what we need to find. We have four answer choices here. A 3d each containing a different combination of angles for this triangle. We're gonna come back to those answer choices as we work through the problem. But one thing we can do right off the bat is look at this triangle and recognize that side R and side Q are equal. They have the same length. What that means is that angle R and angle Q must be equal as well. So we can already eliminate option A and option B just based off of that fact. OK. In those answer, choices, Q and R have different values for the angles. They have to be the same because the side links are the same. All right. So let's try to figure out these angles now that we've eliminated some of these choices and what we can do is start with finding angle P OK. And in order to find angle P, what are we gonna do? Well, we have all three side links. So what we can use is the law of cosine. Now, we're called the law of cosines says C squared is equal to A squared plus B squared minus two A B multiplied by cosine of angle C. In this case, we're gonna rewrite it in terms of Pr and Q angle P is the one we're interested in right now. So side P is going to be the side that's on its own in the equation. So we get P squared is equal to R squared plus Q squared minus T uh sorry minus two multiplied by R multiplied by Q multiplied by cosine of angle P. OK. So the angle inside of our cosine has to match with the side that's on the left hand side of the equation. Now, again, we know all the side lines. The only thing we don't know in this equation is si um angle P. So we'll be able to solve and let's go ahead and isolate cosine of P. We can say that cosine of angle P is gonna be equal to P squared minus R squared minus Q squared, all divided by negative two RQ. And then we can go ahead and substitute in those values. So cosine of angle P will be equal to 11 squared minus seven squared minus seven squared. Divided by negative two, multiplied by seven, multiplied by seven. Hey, if we simplify co sine FP in the numerator, we have 121 minus 49 minus 49 that leaves us with 23. In the denominator, we end up with 49 multiplied by negative two. So we get negative 98. So cosine of P is equal to 23 divided by negative 98. And if we take the inverse cosine, we get that P is going to be equal to 103.6 degrees. And so we have angle peanut. Now we need to figure out angle Q and angle R and there's a couple different ways to do this. Now, one way we could do this is to use the law of signs. OK. But the triangle, we given there's something simpler we can do and we can do that again by recognizing that angle P and angle Q have to be the same. OK? Because the side lengths are the same. Now, we know that the angles in a triangle have to add up to 180 degrees. So angle R polo angle PP angle Q as the equal 180 degrees, we know that angle R is the same as angle Q. So we can write this as Q plus angle P which we just found to be 103.6 degrees plus angle Q is equal to 180 degrees. Subtracting 103.6 from both sides, we get that two, Q must be equal to 76.4 degrees. We can divide both sides by two. When we get that Q must be equal to 38.2 degrees. That tells us that R must be equal to 38.2 degrees because they're the same. So what we're doing is we're essentially saying we know there are 100 and 80 degrees in a triangle. We've already taken up 100 and 3.6 degrees with angle P that leaves 76.4 degrees. And we know it has to be divided evenly between angle Q and R. When we divide by two, we get that angle of 38.2 degrees. Looking at our answer choices. Now, we can see that the correct answer is going to be option CP is 103.6 degrees and Q and R are 38.2 degrees. Thanks everyone for watching. I hope this video helped you in the next one.

Solve each triangle. Approximate values to the nearest tenth.

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Key Concepts

Triangle Classification and Properties

Trigonometric Ratios and Functions

Law of Sines and Law of Cosines

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