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 = 10.8, b = 24.7

Accepted Answer

Hello, everyone. We are asked to find the solution for the right triangle in the given figure. While expressing our angles to the nearest 10th of a degree and rounding lengths to two decimal places. We have a right triangle, QR P with the right angle at angle R, our sides are labeled with lower case letters P Q. And our hypotenuse is R, we are told that side P is 8. inside Q is 80.1. We are given four answer choices with varying values for angles P and Q and side R. I'm gonna start by labeling what we were given. We are told that side P is 8.7 and that side Q is 80.1. I'm gonna label everything I find on our diagram just to help me keep track of what I found and what I haven't found. I'm gonna begin by trying to find the value of the angle at P right now. The angle at P I can see that side P is across from it. So that would be opposite and that side Q is adjacent to it. So opposite over adjacent means we're working with the tangent ratio. So I'm gonna find 10 of P has to equal 8.7 divided by 80.1. So first recall that you need to have your calculator in degree mode. And when we type this into a calculator, remember to use the inverse function. So it's gonna be the tangent inverse function. So with the little negative one exponent and then you can put 8.7 divided by 80.1. And when you do so you should get 6.198. And it just keeps going, we were told to round our angles to the nearest 10th of a degree. So I'm gonna round this to 6.2 degrees. So this means the angle at P is 6.2 degrees. So I might use that to find the rest of my values I'm going to take and try to find the other corner of my triangle. So angle Q I can do this because I know that the sum of the interior angles of a triangle equal 180 degrees. So if the sum of the total of these three angles is 100 80 if I do 180 degrees minus 90 for our right angle minus 6.2 degrees for the angle P, we just found when I type that in, I will get 83.8 and that will be degrees. And this will be the measure of the angle at Q. So the angle at Q is 83.8 degrees. So we have two out of the three things completed, we still need to find side R because this is a right triangle. I'm gonna use the Pythagorean theorem. So recall that the sum of the squares of the legs of a right triangle are equal to the square of the hypotenuse. Or as most people remember it, A squared plus B squared equals C squared. Here, I'm gonna use side P is my A. So 8.7 squared plus side Q is my B so 80.1 squared, this has to equal our unknown side of R squared. So I'm going to do both of those squares. And when I do, we get, I'm gonna actually round. So we have 75. plus 0.1 equals R squared. When I add those two sides together or those two values together, I get 0.70 equals R squared, we're called the opposite of a square as a square root. And when I take the square root of both sides, I am left with 80. equals R. So now I found my hypos, I have 80 point and two decimal places. So 20.57 and now I have all my values. Let's see which answer choice they match. So first they all list P first. So angle at P is 6.2 degrees. So it cannot be answer choice B uh the angle Q is 83.8 degrees. So that so far it looks like it's answer choice D and just verifying my value for R, they have 80.56, I have 80.57, but we may have rounded in different spots. So answer choice D is my 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. a = 10.8, b = 24.7

Key Concepts

Pythagorean Theorem

Trigonometric Ratios (Sine, Cosine, Tangent)

Angle Sum Property of Triangles

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