What are Perpendicular lines

Two intersecting lines will have four angles formed at the intersection points. If all the four angles are equal, then the two lines are said to be perpendicular to each other. We already know by linear postulate theorem that the two vertically opposite angles are equal. Hence if these two lines are perpendicular, then all four angles are 90 degrees.

Examples of perpendicular lines:

In the graph paper, The X-axis and Y-axis are perpendicular.
In an ellipse two axes, minor axis, and major axis are perpendicular.
For a line segment, any shortest line from a point outside the circle is perpendicular.
Tangent and normal to any curve are perpendicular lines.

Slopes of two perpendicular lines:  In coordinate Geometry, when two lines are perpendicular, the product of the slopes of the lines is -1.  This property has a lot of applications in finding the equation of perpendicular lines, length of perpendicular segment from a point to a given line, etc.

For any curve in a graph with equation y = f(x), the slope of the tangent is defined as the rate of change of y with respect to x at that point. The normal to this curve at this point is perpendicular to the tangent line.

Example:  In a circle, with centre at the origin and radius 3, the equation will be of the form (x)²+(y)² = 3². Take any point say (0,3). To find the tangent, we have to find dy/dx.

Differentiating, 2x+2y  =0

Hence, the slope of the normal is perpendicular to x axis or parallel to y axis.

Example for Perpendicular lines from a point to a line

Let AB be a line with coordinates (1,2) and (3,4).  Measure the length of perpendicular line from (-1,1) to this line segment.

We know that the perpendicular line from (-1,1) has a slope  of -1/slope of AB.

Equation of AB is (x-1)/(3-1) = (y-2)/(4-2)  Or x-1 = y-2 Or y = x+1

Slope of AB passing through (1,2) and (3,4) is 4-2/3-1 =1.

Slope of perpendicular line to AB is -1.

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Since the perpendicular line passes through (-1,1) equation of the perpendicular is y-1 = -1(x+1)  or y =-x -1 +1 or y = -x.

To get the foot of the perpendicular line on AB, we solve the two equations by substitution method.

y = x+1 = -x   This on simplification gives 2x=-1 or x=-1/2.

Since y = -x , we have y = +1/2,

So, foot of the altitude from the point (-1,1) is (-1/2,1/2).

The length of the perpendicular segment is between (-1,1) and (-1/2,1/2) is

√[ (-1/2+1)²+(1/2-1)²] = √(1/4+1/4)   =  √(1/2) = 1/1.414 =0.707 approximately.