Understanding the Geometry: Finding Lines at a 45° Angle

In this article, we explore how to find the equations of straight lines that pass through a specific point and make a 45° angle with a given line. This is a common problem in geometry and analytic mathematics, and it is particularly useful in computer science and engineering applications. We will break down the problem into steps and provide the necessary algebraic manipulations to find the desired lines.

Step 1: Determine the Slope of the Given Line

We begin by identifying the slope of the given line, which is represented by the equation:

3x 4y 12

First, we rearrange the equation into the slope-intercept form: y mx b. This gives us:

4y -3x 12

y -(frac{3}{4})x 3

So, the slope (m_1) of the line 3x 4y 12 is (-(frac{3}{4})).

Step 2: Calculate the Slopes of the New Lines

We need to find the slopes of the new lines that make a 45° angle with the given line. The angle (theta) between two lines with slopes (m_1) and (m_2) is given by the formula:

(tan theta left|frac{m_2 - m_1}{1 m_1 m_2}right|)

Given that (theta 45^circ), (tan 45^circ 1). Therefore, we set up the equation:

(1 left|frac{m_2 frac{3}{4}}{1 - frac{3}{4}m_2}right|)

This gives us two cases to consider.

Case 1: Slope (m_2 frac{1}{7})

(frac{m_2 frac{3}{4}}{1 - frac{3}{4}m_2} 1)

(m_2 frac{3}{4} 1 - frac{3}{4}m_2)

(m_2 frac{3}{4} m_2 1 - frac{3}{4})

(frac{7}{4}m_2 frac{1}{4})

(m_2 frac{1}{7})

Case 2: Slope (m_2 -7)

(frac{m_2 frac{3}{4}}{1 - frac{3}{4}m_2} -1)

(m_2 frac{3}{4} -1 frac{3}{4}m_2)

(m_2 - frac{3}{4}m_2 -1 - frac{3}{4})

(frac{1}{4}m_2 -frac{7}{4})

(m_2 -7)

Step 3: Write the Equations of the Lines

We now have two slopes (m_2 frac{1}{7}) and (m_2 -7). We can use the point-slope form of the line equation to find the equations of the lines: [y - y_1 m(x - x_1)]

where the given point is (-1, 4).

For (m_2 frac{1}{7})

[y - 4 frac{1}{7}(x 1)]

Rearranging gives:

[y frac{1}{7}x frac{29}{7}]

For (m_2 -7)

[y - 4 -7(x 1)]

Rearranging gives:

[y -7x - 3]

Final Equations

The equations of the straight lines passing through the point -1, 4 and making a 45° angle with the line 3x 4y 12 are:

(y frac{1}{7}x frac{29}{7}) (y -7x - 3)

Conclusion

This detailed approach demonstrates how to find the equations of lines that meet specific geometric criteria. Understanding the process is essential for solving similar problems and can be applied in various fields such as computer graphics, engineering, and data analysis.