Are Perpendicular Lines Intersecting Lines

Perpendicular lines are formed when ii lines meet each other at the correct angle or 90 degrees. This property of lines is said to be perpendicularity . In this article, we are going to discuss what is perpendicular, the properties of perpendicular lines, the divergence between perpendicular and parallel lines, and many solved examples in detail.

What is Perpendicular?

In Mathematics, a perpendicular is defined as a direct line that makes the right angle (90 degrees) with the other line. In other words, if two lines intersect each other at the right bending, and so the lines are perpendicular to each other. At present, consider the beneath figure. The line PQ and RS intersect each other at right angles and hence we tin can say that the lines PQ and RS are perpendicular to each other.

Perpendicular Lines

What is a Perpendicular Line?

The perpendicular lines are two lines that intersect each other and the angle formed between the two lines should be equal to 90 degrees (right angle). Consider the above-given effigy, the line PQ and RS forms a right bending when the lines intersect at a point. Hence, the lines are perpendicular to each other and mathematically it is represented as PQ⊥ RS.

When a line is perpendicular to the airplane, then it is said to be perpendicular to all the points in the airplane or perpendicular to each line in the plane that information technology intersects. 2 planes are said to exist perpendicular in space if the dihedral angle at which they come across the plane is a right angle.

Notation: Perpendicular lines always intersect at ninety degrees just not all intersecting lines are perpendicular.

Also, read: Intersecting and Non-intersecting Lines

Perpendicular Symbol

Perpendicular lines are represented by the symbol, ' ⊥ '. Suppose, fifty ₁ and l ₂ are 2 lines intersecting each other at ninety degrees, and then they are perpendicular to each other and are represented as l _i ⊥l ₂ . The point of intersection is chosen the foot of the perpendicular .

Properties of Perpendicular Lines

In mathematics, intersecting lines are not always perpendicular lines. The intersecting lines are said to be perpendicular lines if they obey the following properties. The two master properties of perpendicular lines are as follows:

Perpendicular lines ever intersect at the correct angle
If two lines are perpendicular to the same line, then they both are parallel to each other and never intersect.

Gradient of Perpendicular Lines

Suppose two lines AB and CD are perpendicular to each other. The slope of line AB is m₁ and the gradient of other line CD is m₂.

Statement: Two lines are perpendicular to each other if and just if the product of the slope of the two lines equals minus of unity.

Thus, the formula for the gradient of the perpendicular is given as:

In the case of the parallel line, the gradient of the ii lines is parallel to each other.

one thousand_one = m₂

Construction of Perpendicular Lines

Structure of the perpendicular line is a very simple process. The angle between the two lines should be equal to ninety degrees. So to construct perpendicular lines, you lot will need a compass and a straight line ruler or scale. Follow the below steps to describe it:

Draw a horizontal line first.
With the aid of a compass describe an arc at the center of the line say point O, such that it intersects the line at 2 points and at equidistant from O. Let the two points be P and Q.
Again at betoken P and Q, draw the arc inside, such that the two arcs intersect each other at the top and bottom of the horizontal line.
Now join the two-point where the 2 arcs intersect each other.
Now, the line obtained is perpendicular to the horizontal line.

Also, read:

Concurrent Lines
Lines And Angles Class 7
Important Questions Class 9 Maths Chapter half-dozen Lines Angles

Examples

We know that if a ray is rotated near its finish-point, the measure of its rotation is called the bending between its initial and concluding position. The value of any bending is proportional to its amount of rotation and the sense of its rotation. Clearly, the greater the amount of rotation, the larger will be the angle formed. A special example of angles is a right bending, in which the mensurate of rotation of a ray is 90^o. When ii lines or surfaces intersect to form right angles then such lines or surfaces are said to be perpendicular to each other.

Consider the following two-line segments

\(\begin{array}{l}\overline{AB}\end{array} \)

and

\(\begin{array}{l}\overline{CD}\finish{assortment} \)

. These line segments are perpendicular to each other equally they intersect at ninety^o at signal 10. Thus, both the line segments have a common intersection indicate i.e. 10 and are correct angles to each other.

Perpendicular lines 3

Perpendicular lines lie in the same plane i.e. they are co-planar and intersect at correct angles. Thus information technology implies that if you accept two lines that are perpendicular to each other, so these lines will be at right angles and vice versa.

Using but a compass i tin can draw a perpendicular to a line. These straightedge techniques were developed past aboriginal Greeks. In the example of co-ordinate geometry, a line is said to be perpendicular just if the slope of a line has a definite relationship.

Real-Life Examples

If you just await around y'all will find numerous examples of perpendicular lines and surfaces. The corners of the wall intersect each other at right angles, the tiles in the kitchen or the washroom, the intersection of roads at squares, hands of a clock when it strikes exactly iii' O clock, the corners of your desk-bound or the doors are examples illustrating perpendicularity.

What are Parallel Lines?

In contradiction to perpendicular lines, in Geometry, we have parallel lines , which are parallel to each other and practise not meet at whatever point.

Parallel lines

In the above image, the two lines PQ and RS are parallel to each other.

Deviation Betwixt Parallel Lines and Perpendicular Lines

The departure betwixt the parallel lines and perpendicular lines are given below:

Parallel Lines	Perpendicular Lines
Parallel lines are the lines that do not intersect at any signal and the altitude between the 2 lines is always the aforementioned.	Perpendicular lines are the lines that intersect each other at right angles (90 degrees).
The symbol used to represent a parallel line is "\|\|".	The symbol used to represent perpendicular lines is "⊥".
Example of Parallel lines: Opposite sides of a rectangle.	Example of Perpendicular lines: Corner of two walls.

Examples on perpendicular Lines

Example 1:

If i line passes through the points (0, –four) and (–1, –7) and some other line passes through the points (3, 0) and (–3, ii). Are these lines parallel or perpendicular?

Solution:

Slope of starting time line,

chiliad_one = (-seven+4)/-1 = -3/-1 = 3

k₂ = (2-0)/(-3-iii) = 2/(-6) = -⅓

Since, m₁ ≠ m₂, therefore, lines are not parallel.

m_ane.m₂ = 3 ten (-⅓) = -1

Therefore, the two lines are perpendicular.

Example 2:

What is the equation for the line that is perpendicular to 4x−3y=6 through indicate (4,6)?

Solution:

As nosotros know, the slope of perpendicular lines are opposite reciprocals.

Given, 4x−3y=6

Writing in slope-intercept course, we get;

y = (4/3)x – 2

So, slope of the line 4x−3y=6, chiliad = iv/three

Now, the slope of line perpendicular to the given line is (-¾).

Using the coordinates of the bespeak (4,6) and putting the given equation, we become;

half dozen = (-¾) (4) + b

b = 9

Therefore, the required equation of a perpendicular line is given past:

y = (-¾) x + 9

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Frequently Asked Questions on Perpendicular Lines

What are perpendicular lines?

Perpendicular lines are those lines that intersect each other at 90 degrees.

How are parallel lines unlike from perpendicular lines?

Parallel lines are parallel to each other and never intersect. But perpendicular lines intersect at correct angle.

Are all intersecting lines always perpendicular?

Not all intersecting lines are perpendicular to each other. They may intersect at dissimilar angles too.

What is the status for the slope of perpendicular lines?

If the gradient of perpendicular lines are m1 and m2, then;
m_ane.one thousand_two = -1

Observe the slope of a line perpendicular to the line y = –4x + ix.

Given, equation is y = -4x + 9
So, the slope of the given equation is -four.
Now the slope of the lines perpendicular to the given line is:
m = negative reciprocal of the gradient of a given line
1000 = -i/(-4) = i/4