Processing math: 100%

Subscribe to our YouTube channel for the latest videos, updates, and tips.


Angle between Two Straight Lines

We will learn how to find the angle between two straight lines.

The angle θ between the lines having slope m1 and m2 is given by tan θ = ± m2m11+m1m2

Let the equations of the straight lines AB and CD are y = m1 x + c1 and y = m2  x + c2  respectively intersect at a point P and make angles θ1 and θ2 respectively with the positive direction of x-axis.

Let ∠APC = θ is angle between the given lines AB and CD.

Clearly, the slope of the line AB and CD are m1  and m2  respectively.

Then, m1  = tan θ1  and m2  = tan θ2

Now, from the above figure we get, θ2  = θ + θ1  

⇒ θ = θ2  - θ1

Now taking tangent on both sides, we get,

tan θ = tan (θ2  - θ1)

⇒ tan θ = tanθ2tanθ11+tanθ1tanθ2, [Using the formula, tan (A + B) = tanAtanB1+tanAtanB

⇒ tan θ = m2m11+m1m2, [Since, m1  = tan θ1  and m2  = tan θ2]

Therefore, θ = tan1m2m11+m1m2

Again, the angle between the lines AB and CD be ∠APD = π - θ since ∠APC = θ

Therefore, tan ∠APD = tan (π - θ) = - tan θ = - m2m11+m1m2

Therefore, the angle θ between the lines AB and CD is given by,

tan θ = ± m2m11+m1m2

⇒ θ =  tan1m2m11+m1m2


Notes:    

(i) The angle between the lines AB and CD is acute or obtuse according as the value of m2m11+m1m2 is positive or negative.

(ii) The angle between two intersecting straight lines means the measure of the acute angle between the lines.

(iii) The formula tan θ = ± m2m11+m1m2 cannot be used to find the angle between the lines AB and CD, if AB or CD is parallel to y-axis. Since the slope of the line parallel to y-axis is indeterminate.

 

Solved examples to find the angle between two given straight lines:

1. If A (-2, 1), B (2, 3) and C (-2, -4) are three points, fine the angle between the straight lines AB and BC.

Solution:

Let the slope of the line AB and BC are m1 and m2 respectively.

Then,

m1 = 312(2) = 24= ½ and

m2 = 4322= 74

Let θ be the angle between AB and BC. Then,

tan θ = |m2m11+m1m2| = |74121+7412| = |108158|= ±23.

⇒ θ = tan1(23), which is the required angle.

 

2. Find the acute angle between the lines 7x - 4y = 0 and 3x - 11y + 5 = 0.

Solution:  

First we need to find the slope of both the lines.

7x - 4y = 0     

⇒ y = 74x

Therefore, the slope of the line 7x - 4y = 0 is 74

Again, 3x - 11y + 5 = 0     

⇒ y = 311x + 511

Therefore, the slope of the line 3x - 11y + 5 = 0 is = 311

Now, let the angle between the given lines 7x - 4y = 0 and 3x - 11y + 5 = 0 is θ

Now,

tan θ = | m2m11+m1m2| = ±743111+74311 = ± 1

Since θ is acute, hence we take, tan θ = 1 = tan 45°

Therefore, θ = 45°

Therefore, the required acute angle between the given lines is 45°.

 The Straight Line




11 and 12 Grade Math 

From Angle between Two Straight Lines to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.




Share this page: What’s this?

Recent Articles

  1. Expanded form of Decimal Fractions |How to Write a Decimal in Expanded

    May 07, 25 01:48 AM

    Expanded form of Decimal
    Decimal numbers can be expressed in expanded form using the place-value chart. In expanded form of decimal fractions we will learn how to read and write the decimal numbers. Note: When a decimal is mi…

    Read More

  2. Dividing Decimals Word Problems Worksheet | Answers |Decimals Division

    May 07, 25 01:33 AM

    Dividing Decimals Word Problems Worksheet
    In dividing decimals word problems worksheet we will get different types of problems on decimals division word problems, dividing a decimal by a whole number, dividing a decimals and dividing a decima…

    Read More

  3. How to Divide Decimals? | Dividing Decimals by Decimals | Examples

    May 06, 25 01:23 AM

    Dividing a Decimal by a Whole Number
    Dividing Decimals by Decimals I. Dividing a Decimal by a Whole Number: II. Dividing a Decimal by another Decimal: If the dividend and divisor are both decimal numbers, we multiply both the numbers by…

    Read More

  4. Multiplying Decimal by a Whole Number | Step-by-step Explanation|Video

    May 06, 25 12:01 AM

    Multiplying decimal by a whole number is just same like multiply as usual. How to multiply a decimal by a whole number? To multiply a decimal by a whole number follow the below steps

    Read More

  5. Word Problems on Decimals | Decimal Word Problems | Decimal Home Work

    May 05, 25 01:27 AM

    Word problems on decimals are solved here step by step. The product of two numbers is 42.63. If one number is 2.1, find the other. Solution: Product of two numbers = 42.63 One number = 2.1

    Read More