Introduction to Straight Line Equations and Slopes

In the realm of mathematics, the equation of a straight line is a fundamental concept. The standard linear equation is given by y mx b, where m represents the slope of the line, and b is the y-intercept. This equation serves as a powerful tool for analyzing and understanding linear relationships in various applications.

Understanding the Given Problem

The task is to find the equation of a straight line that has an inclination (slope) of 60 degrees and passes through the midpoint of the segment connecting points A (3, 6) and B (0, -8). To approach this, let's break down the problem step by step.

Step 1: Finding the Midpoint

The first step is to determine the midpoint of the segment connecting points A and B. The formula to find the midpoint of a line segment with endpoints (x1, y1) and (x2, y2) is given by: Midpoint ([ (x1 x2)/2, (y1 y2)/2 ]) Applying this formula to points A (3, 6) and B (0, -8), we get:

([ (3 0)/2, (6 - 8)/2 ] [3/2, -2/2] [1.5, -1])

Thus, the midpoint of the segment AB is C (1.5, -1).

Step 2: Calculating the Slope

The next step is to determine the slope of the line. The slope (m) of a line is calculated using the formula: m (y2 - y1) / (x2 - x1) However, since the slope is given as 60 degrees, we need to convert this angle to its tangent value:

m tan(60°) √3 / 1 √3 or about 1.732. For simplicity, we will use the exact value of √3.

Step 3: Using the Line Equation with the Given Point

Now that we know the slope (m √3) and a point on the line (the midpoint C (1.5, -1)), we can use the point-slope form of the equation of a line. The point-slope form is: y - y1 m(x - x1) Substituting the known values:

y - (-1) √3(x - 1.5)

Expanding and simplifying this equation, we get:

y 1 √3x - 1.5√3

y √3x - 1.5√3 - 1

Thus, the equation of the line in the form y mx b is:

y √3x (-1.5√3 - 1)

Conclusion

To summarize, the equation of the straight line with an inclination of 60 degrees and passing through the midpoint of the segment connecting points A (3, 6) and B (0, -8) is given by:

y √3x (-1.5√3 - 1)

This equation accurately describes the linear relationship between the variables and can be used to analyze and predict outcomes based on the given conditions.