In Exercises 39–40, find h to the nearest tenth.

Question

Diagram of triangle PQR with angles 38° and 59°, side SR labeled 288, and height h.

Accepted Answer

Hey, everyone in this problem, we're asked to determine the value of H in the following triangle and to write our answer to the nearest 10th. And so we're given this triangle and what we have is a large exterior triangle with corners P S and Q. And the side H that we're looking for is between P and Q. And we have a right angle at corner Q. And then this triangle is kind of split by this additional line. OK, going from corner P to between S and Q. And we have this new corner R and this splits our original triangle into two additional triangles. We have four answer choices. Option a 424.1. Option B 422.4, option C 441.2 and option D 442.1. So let's think about finding H OK. In the immediate triangle. OK. Pr Q, we have only the angle R, we have no other side length, no other angles. So it's not enough information to find H OK. If we look at this larger triangle again, we have the angle at S but we don't have a full side length, we don't have any other angles. So finding H we're gonna need to find another side length and the side length we're gonna choose to find is gonna be cute. We're gonna call it Q and it's gonna be the line connecting R and P and you could also choose to find S P that would also work. But we're gonna try for Q, OK. If we can find side length Q, then we can use the definition of our trigonometric functions. OK? Because we have this right angle triangle with the angle and the hypotenuse to find H. All right. So we need to find Q. How do we find Q? Well, let's look at our other triangle. OK. So we have this triangle S pr we have this opposite angle to Q. So that's good. OK. Let's think about using our sign law to relate these opposite angles and opposite sides. So we have our angle 38 degrees. We have this side cue we're looking for in order to find that we need the ratio of another side link to angle. So the other side length we're given, we have 288 on side S R. So we can find this opposite angle which we're gonna call alpha, then we'd be able to use our sign law to find side length QE. So how can we find? Awful? Well, let's start on the opposite side. OK. We have this angle at R of 59 degrees. If we continue that angle down, this forms a straight line, we know that must be 180 degrees. So let's call this the and we know that theta is gonna be equal to 180 degrees minus that 59 degrees of the angle at R which is gonna be degrees. Yeah. So we have the data and you might be thinking, how does that help A? We want the angle alpha, we're finding a different angle. But remember that the angle in a tri angles in a triangle all add up to 100 and degrees. So now we have two of the angles we can find the third. OK. Alpha is gonna be equal to that total 180 degrees that we know a triangle has minus the angles. We already know 38 degrees minus 121 degrees. And so our angle alpha is gonna be equal to 21 degrees. All right. So now we can use our law of science, OK. We know that the side length from S to R divided by the sign of the angle opposite that. So sine of the angle alpha, it's gonna be equal to the side length queue that we're looking for divided by sine of the opposite angle which in this case is sine of angle S OK. So that's a lot of signs, we just have this ratio of side length to opposite angle and we can set them equal to each other. Now, filling in what we know we have divided by sign of degrees is equal to Q divided by sine of 38 degrees. And we want a soft for QE. So we get that QE is equal to and multiplying both sides by sine of 38 degrees, 288 multiplied by sine of degrees divided by a sign of 21 degrees. OK. So we have this value of tuna and we're gonna leave it like this, we're gonna try to avoid round off error where we can. So we're gonna leave it with this sign of 38 divided by sine of 21. And we will work out those exact values or the estimated approximate values once we get to the final answer. OK? So this is our side length. You now looking at this triangle pr Q, we have an angle, we have a side length, we know it's a right angle triangle. So we can find our side. H Now we're gonna use a trig definition. OK. And which one we're gonna use well from our angle, we can see that we're relating the opposite side in the hypotenuse. And so we want to use S OK. So sign of our angle, which we're gonna call it the angle R is gonna be equal to the opposite side which is H divided by the hypotenuse, which is cute. This gives us sign of 59 degrees is equal to H divided by all of this mess. OK? And let's actually rewrite this first. So we don't have to divide and do the reciprocal. We can write that H is gonna be equal to sign of the angle R multiplied by Q. OK? That, that's gonna make it a little bit easier to write. It's gonna give the exact same answer. We just simplified a step further. Before substituting in our numbers, we get that sign of 59 degrees multiplied by 288 multiplied by sine of 38 degrees divided by nine of 21 degrees. OK? And now we can go ahead and work this out on our calculator. OK. The question was asking us to round to the nearest 10th. We're gonna do that and we get that H is approximately equal to 424.1. So using the law of science and using the definition of the sign function, we were able to find that H is equal to 424. which corresponds with answer choice. A thanks everyone for watching. I hope this video helped see you in the next one.

In Exercises 39–40, find h to the nearest tenth.

Key Concepts

Trigonometric Ratios

Height in Right Triangles

Angle Sum Property

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