Use the result from Exercise 80 to find the acute angle betwee...

Question

Use the result from Exercise 80 to find the acute angle between each pair of lines. (Note that the tangent of the angle will be positive.) Use a calculator, and round to the nearest tenth of a degree.

x + y = 9, 2x + y = -1

Accepted Answer

Hello, everyone. We are asked to determine the acute angle between the pair of lines given below. We will round our answer to the nearest 10th of a degree. The lines we are given are two X minus three, Y equals eight and three, X minus two, Y equals 14. Our answer choices are A 22.6 degrees B 42.6 degrees C 67.4 degrees D 55.3 degrees. First, I recall that we have a formula that will help us with this. But we first need to know the slope of our lines. So I'm going to start with the first line two, X minus three, Y equals eight. And I want to find it slope, which is M, this is currently set up in standard form, which is A plus B equals C. And in that format, the slope is the opposite of A divided by B. So in this example, A is two, so the opposite of A would be negative two B would be negative three. So the slope of this first line which I'm gonna call M sub one is two thirds. The next line three. X minus two, Y equals 14 again is in standard form. So A X plus by equals C format. So we could find the slope the same way M will equal the opposite of A divided by B. Here A is three because that's the value next to X. So the opposite of three is negative three divided by B which is negative two. So our slope here, M sub two is three halves. So now that we know M sub one and M sub two, we can go find the formula that'll help us find the angle. And in this case, it's gonna be the tangent of alpha. I'm sorry, the tangent of theta equals the absolute value of M sub two minus M sub one. So the slopes divided by one plus M sub one multiplied by M sub two. So we're going to find the difference of our slopes and then we're dividing by one plus the product of our slopes in an absolute value. So the tangent of theta will equal the absolute value of M sub two. So three halves minus M sub one which is two thirds divided by one plus two thirds multiplied by three halves. So simplifying this will have the absolute value and in the numerator, I'll have a common denominator of six. So to get the first fraction to have a denominator of six, I need to multiply numerator and denominator by three. So I have nine sixths minus I need to multiply numerator and denominator by two for my second fraction. So 4/6 in the denominator, I'll have one plus and then multiplying the fractions together, I get 66, which is the same as one. So I'll have the absolute value of 5/6 divided by one plus one which is two. And I'm going to rewrite this horizontally. So I have the absolute value a 56 divided by two. It will be the absolute value of 5/6 multiplied by one half because that's the reciprocal of two. And this will equal the absolute value of 5/12 which is 5/12. So we can say that the tangent of theta equals 5/12. But we want to know what is that angle? So we are going to have to work backwards. I'm gonna use a calculator and in my calculator first, I'm gonna make sure it is in degree mode or this will not work next. I'm gonna look for the button that says tangent raise the negative first power and then I'm entering five divided by 12. And when I do, I get 22.6199 since we want this rounded to the nearest 10th of a degree, this would round to 22.6 degrees. So this would be the angle but formed between these two lines. And that's the answer choice. A 22.6

Use the result from Exercise 80 to find the acute angle between each pair of lines. (Note that the tangent of the angle will be positive.) Use a calculator, and round to the nearest tenth of a degree.
x + y = 9, 2x + y = -1

Key Concepts

Slope of a Line

Tangent of an Angle

Acute Angle

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