Mathematics

Perpendicular lines Representation, Properties, and other details

When two lines meet each other at an angle of 90 degrees, they are said to be perpendicular lines.

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                                                                                                                        Image is taken from cuemath

As given in the figure, there are two lines, AB and CD, intersecting each other at an angle of 90 degrees.

Representation

The lines which intersect each other are represented by “⊥” (AB ⊥ CD).

The point where these two lines meet is called the Foot of the perpendicular.

It is donated by a square shape at the point.

Properties

Perpendicular lines always intersect each other but all are not perpendicular to each other, if it makes “L” shaped, the angle is right angle so the lines are perpendicular.

Some of the properties of perpendicular lines are as follows:

  • Perpendicular lines always intersect or meet each other at a right angle.
  • Perpendicular lines always make an angle of 90 degrees.
  • Two lines which do not intersect each other cannot be perpendicular, they are parallel.
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                                                        Image is taken from splashlearn

  • Two lines which intersect each other at acute angles are not perpendicular.
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                                                    Image is taken from splashlearn 

  • Any line which is not intersecting at right angles is not perpendicular.
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                                                      Image is taken from splashlearn

Shapes

Some of the common mathematical shapes have perpendicular lines:

  • Square
  • Right-Angled Triangle
  • Rectangle

The slope of Perpendicular Lines

AB and CD are the two lines that make an angle of 90° and are said to be perpendicular.

The slope of AB is m1 and the Slope of CD is m2.

If the product of the slope of the two lines, AB and CD equal to -1 as given below:

m1*m2 = -1

then they are said to be perpendicular.

Perpendicular lines Construction

Take a scale (ruler) and draw 2 lines that have an angle of 90 degrees (straight line).

As we can see in the figure below:

Perpendicular lines can be drawn not only from a scale but also using a compass and a protractor.

  • Using a Protractor

A protractor is an important measuring instrument in math and it is used in measuring angles and also making perpendicular lines.

Below are a few simple steps to construct perpendicular lines.

Step 1: Put the bottom end of the protractor (where you can see a DOT) at point P.

Step 2: Make a point B at 90 degrees on the Protractor

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                                                                                                                     Image is taken from cuemath

Step 3: After removing the protractor, using a scale join the point B to Point P.

It will make a line as in the below figure:

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                                                                                                               Image is taken from cuemath

Example

In the given figure, the lines AB and CD are perpendicular (⊥) to each other.

If the given BOC angle is equal to 90 degrees, find the value of x

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                                                                                                                    Image is taken from cuemath 

Solution:

First things first, a right-angled triangle can be made or given in any x,y plane, don’t be alarmed if you see it upside down.

BOC = 90°

x + 63° = 90°

x = 90° – 63°

x = 27°

Where x is 27°, if we add both 27° and 63° we will get an angle of 90° which suggests that the lines are perpendicular to each other.

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