Solve the right triangle shown in the figure. Round lengths to...

Question

Solve the right triangle shown in the figure. Round lengths to two decimal places and express angles to the nearest tenth of a degree. B = 16.8°, b = 30.5

Right triangle ABC with angle B 16.8°, side b 30.5, right angle at C, sides a and c labeled.

Accepted Answer

Hello, everyone. We are asked to find the solution for the right angle in the given figure, expressing angles to the nearest 10th of a degree and rounding lengths to two decimal places. We are given a right triangle. QR P with the right angle at angle R, our sides are labeled with lowercase letters PQ are our legs. R is our hypotenuse. We are now told that the angle at Q measures 7.3 degrees inside Q measures 9.7. We are given four answer choices, all of which vary slightly in the value for the angle at P side P and side R. First, I'm gonna label what we are given. So we are told that the angle at Q is 7.3 degrees and we are told that sign Q is 9.7. So I put that on my diagram before I do anything else. Next, I'm gonna find the angle at P. I'm gonna do this by knowing that the interior angles of a triangle total to 180 degrees. So 180 degrees minus angle R which is the right angle, 90 degrees minus 7.3 degrees for Q, when I do that subtraction, we get that the angle at P must be 82.7 degrees. So we know this, I'm gonna label it on my diagram. The angle at P is 82.7 degrees from there. I look at what sides I have, I know side cut in relation to angle Q that is the opposite. That might not be the easiest trig ratio to work with. However, in relation to angle P, that would be the adjacent side. So if I used the tangent of the value of angle P, so 82.7 degrees, I can say this must equal the opposite side or lowercase P divided by the adjacent side of Q which is 9.7. This will be easier math to do than if I had worked off of angle Q, make sure your calculator is in degree mode. And then the tangent of 82.7 degrees is 7.80622. It does continue going. But that is where I rounded it and that'll equal P divided by 9.7. I must multiply both sides by 9.7 to get the P by itself. And when I do so I get that P equals 75.7203. But as we want this rounded to two decimal places, this will be rounded to 75.72. So the value of P is 75.72. I'm gonna put that on my diagram. Also. Next, I still need to find the value of R I could either use another trig ratio or the Pythagorean theorem. I'm gonna use the Pythagorean theorem. Recall that the Pythagorean theorem states that the sum of the squares of the legs of a right triangle is equal to the square of its hypotenuse or A squared plus B squared equals C squared. I'm gonna use my P value as A. So 75.72 squared plus Q will be my B side. So 9.7 squared equals my unknown side of R squared. So 75.72 squared is 5733.5184. And then 9.7 squared is 94.09. And this equals R squared. Still combining these, I get 5827.6084 equals R squared. The opposite of a square is a square root. So I'm gonna square root both sides. I'm gonna get R equals 76.33877. Again, we want our sides rounded to two decimal places. So this would be that R equals 76.34. I again, I'm gonna put that on my diagram and now I have all my values on my diagram. So I'm going to compare it to my answer choices. My angle at P has to be 82.7 degrees. So that narrows it down to answer choices B or C for the side P I have it as 75.72. So that matches answer choice C now I'm just double checking that R also matches answer choice C and it does, the R is 76.34. So answer choice C is our answer. Have a nice day.

Solve the right triangle shown in the figure. Round lengths to two decimal places and express angles to the nearest tenth of a degree. B = 16.8°, b = 30.5

Key Concepts

Right Triangle Properties

Trigonometric Ratios (Sine, Cosine, Tangent)

Solving Right Triangles

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