Solve each problem. (Source for Exercises 49 and 50: Parker, M., ...

Question

Solve each problem. (Source for Exercises 49 and 50: Parker, M., Editor, She Does Math, Mathematical Association of America.)

Find a formula for h in terms of k, A, and B. Assume A < B.

Accepted Answer

Hey, everyone in this problem, we're asked to derive a mathematical formula for X in terms of YD and D for the following triangle. We're told to note that angle C is less than angle D. So we're giving triangle P QR, OK. And this is broken into two smaller triangles with this line passing through the middle of this triangle forming this additional corner S OK, inside X that we're looking for is between corners P and S. Now we're given four answer choices, options, A through D and each of them are just a different expression involving Y and then tangent of C and tangent of D, the two angles we are told some information about and we're gonna come back to these answer choices once we work through this problem. So we're looking for this side blank X hm. And let's think about how we can kind of make up this value effects. OK. We can do two things. We can find the total length from P to Q. We're gonna call that A can we get the length from S to you? So we're gonna call beat. Now, if we do that, we can write X which we're looking for as A minus B and we're just taking that total length, subtracting the one portion of it to get that remaining portion. Now, why do we want to do that? Well, what you notice is that if we look at just the triangle S QR and we ignore this top part, I mean, we can relate this side B that we've written to the side Y that we know and the angle C that we know or that we have notation for similarly, we can relate side A if we look at the big triangle P QR to angle B and side Y. OK. So this is gonna give us a way to write A and B in terms of the variables we want in our expression YC and D and therefore write X in terms of those variables as well. All right. So let's start with finding what A is. So we look at this triangle P QR, OK. Now we want to relate the two sidelines, OK? That are not the hypothesis. And so what we're gonna do is we're gonna use the tangent, OK. We have a 90 degree angle. So we can use our tangent can recall the tangent of an angle theta is going to be equal to the opposite side divided by the adjacent side. So in this case, we can say that tangent of D is going to be equal to A divided by Y. And so A is going to be equal to Y multiplied by tangent of D. OK. So now we've written a, in terms of the variables we want Y and D, we're gonna do the same for B and for B, we're gonna use the exact same expression tangent of data is equal to the opposite side divided by the adjacent side. But now we're looking at this smaller bottom triangle. So now our angle is C, so we have a tangent of C. This is gonna be equal to the opposite side which is B divided by the adjacent side, which is again, Y, so we can write that B is equal to Y multiplied by tangent of C. OK. So again, we've written this value of B in terms of the variables of interest Y and C and now we can put this all together into our expression for X. OK. So now we can write X as Y multiplied by tangent D minus Y multiplied by tangent of C mode is both terms that we have in our expression for X have this Y in them. So we can factor that out and we can write X is equal to Y weli by tangent of D minus tangent of C. OK. And that's gonna be that final expression for side link X in terms of YC and D. If we compare this to our answer choices, OK, let's just go through each of them. For option A, we can eliminate this rate off the bat, we have Y divided by an expression involving tangent. So that's not what we have. In our case, we're multiplying. So we can eliminate option A. In option B, we're adding tangent of D and tangent of C inside of those brackets. We don't want to do that. So we can eliminate option B in option C. This expression looks really similar. However, we have tangent of C minus tangent of D in the brackets instead of the other way around. So we can eliminate that one. And option D is the correct answer. We have X is equal to Y multiplied by tangent of D minus tangent of C. Thanks everyone for watching. I hope this video helped see you in the next one.

Solve each problem. (Source for Exercises 49 and 50: Parker, M., Editor, She Does Math, Mathematical Association of America.) Find a formula for h in terms of k, A, and B. Assume A < B.

Key Concepts

Trigonometric Functions

Algebraic Manipulation

Inequalities

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