The Angle Between Two Lines

Oct 13, 2025 by magicalceylontours.com 56 views

Introduction

In mathematics, finding the equation of a line that passes through a given point and makes a specific angle with another line is a fundamental problem in geometry and trigonometry. This problem involves using the concept of the slope of a line and the angle between two lines to determine the equation of the desired line. In this article, we will discuss how to find the equation of the lines through a given point and making an angle of 45° with a straight line.

The Slope of a Line

The slope of a line is a measure of how steep it is. It is defined as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. The slope of a line can be calculated using the formula:

m = (y2 - y1) / (x2 - x1)

where m is the slope of the line, and (x1, y1) and (x2, y2) are two points on the line.

The Angle Between Two Lines

The angle between two lines can be calculated using the formula:

tan(θ) = |m1 - m2| / (1 + m1m2)

where θ is the angle between the two lines, and m1 and m2 are the slopes of the two lines.

Finding the Equation of the Lines

To find the equation of the lines through a given point and making an angle of 45° with a straight line, we need to follow these steps:

Find the slope of the given straight line.
Find the slope of the desired line using the formula for the angle between two lines.
Use the point-slope form of a line to find the equation of the desired line.

Step 1: Find the Slope of the Given Straight Line

The given straight line is 2x - 3y + 7 = 0. We can rewrite this equation in the slope-intercept form (y = mx + b), where m is the slope of the line. To do this, we need to isolate y on one side of the equation.

2x - 3y + 7 = 0 -3y = -2x - 7 y = (2/3)x + 7/3

The slope of the given straight line is 2/3.

Step 2: Find the Slope of the Desired Line

The desired line makes an angle of 45° with the given straight line. We can use the formula for the angle between two lines to find the slope of the desired line.

tan(45°) = |m1 - m2| / (1 + m1m2) 1 = |(2/3) - m2| / (1 + (2/3)m2)

To solve for m2, we can multiply both sides of the equation by (1 + (2/3)m2).

1 + (2/3)m2 = |(2/3) - m2|

Now, we can expand the absolute value on the right-hand side of the equation.

1 + (2/3)m2 = (2/3) - m2

Next, we can add m2 to both sides of the equation.

1 + (5/3)m2 = (2/3)

Now, we can subtract 1 from both sides of the equation.

(5/3)m2 = (2/3) - 1 (5/3)m2 = (-1/3)

Finally, we can multiply both sides of the equation by (3/5) to solve for m2.

m2 = (-1/3) * (3/5) m2 = -1/5

The slope of the desired line is -1/5.

Step 3: Find the Equation of the Desired Line

We can use the point-slope form of a line to find the equation of the desired line. The point-slope form of a line is:

y - y1 = m(x - x1)

where (x1, y1) is a point on the line, and m is the slope of the line.

We are given the point (2, -3), and we have found the slope of the desired line to be -1/5. We can substitute these values into the point-slope form of a line.

y - (-3) = (-1/5)(x - 2)

Now, we can simplify the equation.

y + 3 = (-1/5)(x - 2)

Next, we can multiply both sides of the equation by 5 to eliminate the fraction.

5(y + 3) = -x + 2

Now, we can expand the left-hand side of the equation.

5y + 15 = -x + 2

Next, we can add x to both sides of the equation.

5y + x + 15 = 2

Finally, we can subtract 15 from both sides of the equation.

5y + x = -13

The equation of the desired line is 5y + x = -13.

Conclusion

In this article, we have discussed how to find the equation of the lines through a given point and making an angle of 45° with a straight line. We have used the concept of the slope of a line and the angle between two lines to determine the equation of the desired line. We have followed the steps of finding the slope of the given straight line, finding the slope of the desired line, and using the point-slope form of a line to find the equation of the desired line. The equation of the desired line is 5y + x = -13.

References

[1] "Slope of a Line." Math Open Reference, mathopenref.com/slope.html.
[2] "Angle Between Two Lines." Math Open Reference, mathopenref.com/anglebetweenlines.html.
[3] "Point-Slope Form of a Line." Math Open Reference, mathopenref.com/point-slopeform.html.

Introduction

In our previous article, we discussed how to find the equation of the lines through a given point and making an angle of 45° with a straight line. In this article, we will answer some of the most frequently asked questions related to this topic.

Q1: What is the slope of the given straight line?

A1: The slope of the given straight line is 2/3. This can be found by rewriting the equation of the line in the slope-intercept form (y = mx + b), where m is the slope of the line.

Q2: How do I find the slope of the desired line?

A2: To find the slope of the desired line, you can use the formula for the angle between two lines:

tan(θ) = |m1 - m2| / (1 + m1m2)

where θ is the angle between the two lines, and m1 and m2 are the slopes of the two lines. In this case, the angle between the two lines is 45°, and the slope of the given straight line is 2/3.

Q3: What is the equation of the desired line?

A3: The equation of the desired line is 5y + x = -13. This can be found by using the point-slope form of a line (y - y1 = m(x - x1)), where (x1, y1) is a point on the line, and m is the slope of the line.

Q4: Can I use a different angle instead of 45°?

A4: Yes, you can use a different angle instead of 45°. However, you will need to use the formula for the angle between two lines to find the slope of the desired line.

Q5: How do I find the equation of the line if I don't know the slope of the given straight line?

A5: If you don't know the slope of the given straight line, you can find it by rewriting the equation of the line in the slope-intercept form (y = mx + b), where m is the slope of the line.

Q6: Can I use this method to find the equation of the line if the given point is not on the line?

A6: No, this method will not work if the given point is not on the line. You will need to use a different method to find the equation of the line.

Q7: How do I find the equation of the line if the given straight line is not in the slope-intercept form?

A7: If the given straight line is not in the slope-intercept form, you can rewrite it in the slope-intercept form (y = mx + b), where m is the slope of the line.

Q8: Can I use this method to find the equation of the line if the angle between the two lines is not 45°?

A8: Yes, you can use this method to find the equation of the line if the angle between the two lines is not 45°. However, you will need to use the formula for the angle between two lines to find the slope of the desired line.

Q9: How do I find the equation of the line if I don't know the equation of the given straight line?

A9: If you don't know the equation of the given straight line, you can find it by using the slope-intercept form (y = mx + b), where m is the slope of the line.

Q10: Can I use this method to find the equation of the line if the given point is not a point on the line?

A10: No, this method will not work if the given point is not a point on the line. You will need to use a different method to find the equation of the line.

Conclusion

In this article, we have answered some of the most frequently asked questions related to finding the equation of the lines through a given point and making an angle of 45° with a straight line. We hope that this article has been helpful in clarifying any doubts you may have had.

References

[1] "Slope of a Line." Math Open Reference, mathopenref.com/slope.html.
[2] "Angle Between Two Lines." Math Open Reference, mathopenref.com/anglebetweenlines.html.
[3] "Point-Slope Form of a Line." Math Open Reference, mathopenref.com/point-slopeform.html.