# Perpendicular Lines and its Application

Perpendicular lines are two intersecting lines that meet at right angle or exactly 90°. Let’s bring that line now to coordinate plane. In Analytical Geometry, two lines are perpendicular if the product of their slopes is equal to -1. This time we will practice how to solve problems involving these lines.

**Sample Problem 1: Equation of another line perpendicular to the given line**

Find the equation of the line through and perpendicular to

**Solution:**

The product of the slopes of two equations is equal to -1.

Assuming that m_{1} is the slope of given line and m_{2} is the slope of the required line we establish the relationship below.

.

Solve for m_{1}:

in ,

m is the slope the line. It is clear then that m_{1}=2

Solve for m_{2}:

Generate equation of the line:

Since we have the given point (1,2) and slope -1/2

the latter is the required equation.

**Sample problem 2: Equation of perpendicular bisector**

Find the equation of perpendicular bisector of the segment joining (2,1) and (-4,7).

**Solution:**

Solve for slope of the segment:

Treating m as m_{1}. It is obvious that m_{2}=1 for their product become -1.

Generating equation of bisector:

Since this is a bisector it will divide the segment into two equal parts and will pass through the midpoint

Let be the coordinate of the midpoint

where x_{1} and x_{2} are the x- coordinates of the segment given.

For y_{m}:

where y_{1} and y_{2} are the y-coordinates of the segment given

We already have the slope at the same time the point we will be able to determine the equation of the bisector right now.

, y_{m}=y_{1} and x_{m}=x_{1}

this is the required equation

**Sample Problem 3: Point Reflection**

The point (-1,-2) is reflected to 1^{st} quadrant with respect to the line x+y=0. What is the point of reflection?

Solution:

Draft the point and the line in your paper like this.

From A(-1,-2) draw a line perpendicular to x+y=0. Draft the required point. Let O is the

intersection of the lines. And d is the distance from point A to O. Then d is also the distance

from O to B.

Solve the equation of the line through (-1,-2) and perpendicular to x+y=0

Slope of given equation is -1. So the slope of perpendicular line is 1.

Solve for intersection of the line:

x+y=0 – eqn.1

x-y-1=0 – eqn.2

Solving the equations simultaneously the system is (1/2, -1/2)

The intersection or point O is the midpoint of the line.

From the midpoint formula,

Solve for x_{1}:

Solve for y_{1}:

The required point is **(2,1)**.

Practice Problems:

1. Find the equation of the line through (0,1) and perpendicular to 3x-2y=1.

2. Find the equation of the line through (2,-7) and perpendicular to y-axis.

3. Find the equation of perpendicular bisector of the (3,2) and (-1,6).

4. The point (3,2) is reflected with respect to y-axis. What is the point of reflection?

5. The point (2,-4) is reflected with respect to equation 3x+y-3=0. What is the point of reflection?

Answer key will be available soon.

### Dan

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