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. A = 23.5°, b = 10

Accepted Answer

Hello, everyone. We are asked to find the solution for the right triangle. 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 lowercase P. QR with R being our hypotenuse. We are given that the angle at P is 13. degrees. And that side Q measures 12, we have four answer choices with varying numbers for angle Q sides, P and R. I'm gonna start by labeling what we are given. So we know that side Q equals 12. We know that the angle at P is 13.6 degrees and we need to find angle at Q side P and side R. So I'm gonna start by finding the angle that's at Q, I recall that the internal angles of a triangle total 100 80 degrees. So this means I can work backwards. So I'm gonna start with 100 80 degrees, subtract the right angle which is 90 degrees and then subtract the angle at P which is 13.6 degrees. And when I do that subtraction. I find that the angle at Q must be 76.4 degrees. So Q is gonna equal 76.4 degrees. And I'm gonna label that on my diagram also. So we found one of our values. Let's find another. So we can either find the side P or the hypotenuse. I'm going to probably find side P first. So side P is a cross or opposite angle P and I know Q which is adjacent. So opposite and adjacent lead me to think that I wanna use tangent. So I wanna find side P and I can say that the tangent of angle P is the opposite side divided by the adjacent side. So the tangent of 13.6 degrees equals the opposite side which we don't know. So P divided by the adjacent side which we do know which is 12. So I'm gonna use a calculator making shorts in degree mode to find out what the tangent of 13.6 degrees is. And I get that the tangent of 13.6 degrees is 0.2419. And that's gonna equal P divided by 12. So to get the P by itself, we'll divide or I'm sorry, we're gonna multiply both sides by 12. So that way it cancels out the division. And when I do so I get that P lowercase P would have to equal 2. our directions say we want lengths rounded to two decimal places. So that means side P would be a length of 2.90. So I'm gonna mark that down on my diagram also. That way I have everything I need in one place when I go to look for my answers and I now need to find side R I could use another trig function or I could use the Pythagorean theorem because I know this is a right triangle. I'm gonna use the Pythagorean theorem. So A squared plus B squared equals C squared, R is our hypotenuse. So R is gonna be like C so we are gonna have our two legs squared. So 12 squared plus 2. squared equals R squared. 12 squared is 144 plus 2.9 squared is 8.41 that will equal R squared. When I add those together, I get 152. equals R squared. The opposite of a square is a square root. So taking the square root of both sides, we will get that R equals 12.3454. Again, we want sides rounded to two decimal places. So R would equal 12.35. I'm gonna label that on my diagram also. And now we have all the information we need to check our answer choices. So starting with angle Q, that means it could be answer choice BC or D. So let's keep going. Side P being 2.90. It still could be, oh, it can't be B I'm, I apologize. Q is not the same. B so it has to be answer choice C or D and matching R, it has to be answer choice. C so D is Q is 76.4 degrees, P is 2.90 and R equals 12.35. 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. A = 23.5°, b = 10

Key Concepts

Right Triangle Properties

Trigonometric Ratios (Sine, Cosine, Tangent)

Angle and Side Relationship in Right Triangles

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