How to Find the Equation of a Straight Line Making an Angle of 30° with the X-Axis and Passing Through the Midpoint of the Line Joining (-23, 85)

Identifying the equation of a line based on specific angle and midpoint can be a practical skill for various applications, including geometry, engineering, and data analysis. This article will guide you through the process step-by-step, helping you understand the underlying mathematical concepts.

Step-by-Step Guide

Step 1: Find the Midpoint of the Line Segment

To begin, we need to determine the midpoint of the line segment joining the points (-2, 3) and (8, 5). The midpoint formula for two points (x?, y?) and (x?, y?) is:

Midpoint Formula: ((x_m, y_m) left(frac{x_1 x_2}{2}, frac{y_1 y_2}{2}right))

Substituting the given points:

x_m frac{-2 8}{2} frac{6}{2} 3

y_m frac{3 5}{2} frac{8}{2} 4

Thus, the midpoint is (3, 4).

Step 2: Determine the Slope of the Line

The slope (m) of a line that makes an angle (θ) with the x-axis is given by:

Slope Formula: (m tan(theta))

For θ 30°, we have:

m tan(30°) frac{1}{sqrt{3}} approx 0.577

Step 3: Use the Point-Slope Form of the Line Equation

The point-slope form of the line equation is:

Point-Slope Form: (y - y_1 m(x - x_1))

Using the midpoint (3, 4) and the slope (m frac{1}{sqrt{3}}), the equation becomes:

y - 4 frac{1}{sqrt{3}}(x - 3))

Step 4: Rearrange to Slope-Intercept Form

To convert the equation into the slope-intercept form (y mx b), we need to simplify:

y - 4 frac{1}{sqrt{3}}x - frac{3}{sqrt{3}})

Combining terms, we get:

y frac{1}{sqrt{3}}x - frac{3}{sqrt{3}} 4

Since (frac{3}{sqrt{3}} sqrt{3}), the equation simplifies to:

y frac{1}{sqrt{3}}x - sqrt{3} 4)

This can be written as:

y frac{1}{sqrt{3}}x 4 - sqrt{3})

Final Equation

The equation of the line in slope-intercept form is:

Final Equation: (y frac{1}{sqrt{3}}x 4 - sqrt{3})

Application and Practical Usage

This method is crucial for several practical applications, such as:

Engineering and Architecture: Determining the alignment of structures. Data Analysis: Understanding trends and relationships in datasets. Navigation and Mapping: Calculating paths and intercepts.

By mastering the steps involved in finding the equation of a straight line, you can effectively apply these techniques in diverse fields.