How to Find the Equation of a Line Perpendicular to a Segment and Passing Through a Point

In geometry and algebra, understanding the relationship between lines and segments can be quite useful. One such problem involves finding the equation of a line that is perpendicular to a given segment and passes through a specific point. In this guide, we will break down the process step-by-step to solve this problem.

Identifying the Segment and the Point

Let's start with the given points:

A(-4, -1) B(2, 2)

We need to find the equation of a line that is perpendicular to the line segment joining these two points and passes through the point that is 1/3 of the way from A to B.

Step 1: Finding the Coordinates of the Point

Division Ratio

The point that divides the segment AB in the ratio 1:2 can be found using the section formula. The formula for a point P that divides AB in the ratio m:n is:

P left( frac{mx_2 nx_1}{m n}, frac{my_2 ny_1}{m n} right)

Application of the Formula

Given:

A(-4, -1) B(2, 2) Ratio m:n 1:2

Plugging in the values:

P left( frac{1(2) 2(-4)}{1 2}, frac{1(2) 2(-1)}{1 2} right)

Calculations:

x frac{2 - 8}{3} -2

y frac{2 - 2}{3} 0

Therefore, the point P is at (-2, 0).

Step 2: Determining the Slope of the Line Segment

The slope of the line segment AB is given by:

m_{AB} frac{y_2 - y_1}{x_2 - x_1} frac{2 - (-1)}{2 - (-4)} frac{3}{6} frac{1}{2}

Step 3: Finding the Slope of the Perpendicular Line

The slope of a line perpendicular to another line is the negative reciprocal of the slope of that line. Thus:

m_{perp} -frac{1}{m_{AB}} -frac{1}{frac{1}{2}} -2

Step 4: Using the Point-Slope Form to Find the Equation of the Line

Point-Slope Form of the Line Equation

The general form of the point-slope equation is:

y - y_1 m(x - x_1)

Applying the Point and Slope

Using point P(-2, 0) and the slope -2:

y - 0 -2(x 2)

Simplifying:

y -2x - 4

Final Result

The equation of the line that is perpendicular to the segment joining A(-4, -1) and B(2, 2) and passes through the point P(-2, 0) is:

" "boxed{y -2x - 4}" "

By breaking down the problem into smaller steps and using the appropriate formulas, we can easily find the desired equation of a line in geometry and algebra. This process can be applied to other similar problems involving line equations and perpendicular segments.