PROPERTIES OF PARALLEL AND PERPENDICULAR LINES

Properties of Parallel Lines

Property 1 :

Let m₁ and m₂ be the slopes of two lines.

If the two lines are parallel, then their slopes will be equal.

m₁ = m₂

Property 2 :

Let us consider the general form of equation of a straight line.

ax + by + c = 0

If the two lines are parallel, then their general forms of equations will differ only in the constant term and they will have the same coefficients of x and y.

ax + by + c₁ = 0

ax + by + c₂ = 0

Property 3 :

Let us consider the slope intercept form of equation of a straight line.

y = mx + b

If the two lines are parallel, then their slope-intercept form equations will will differ only in the "y"- intercept.

y = mx + b₁

 y = mx + b₂ 

Property 4 :

Let l₁ and l₂ be two lines.

If the two lines are parallel, the angle between them and the positive side of x-axis will be equal.

The figure given below illustrates the above situation.

Property 5 :

If the two lines are parallel, the perpendicular distance between them will be same at everywhere.

The figure given below illustrates the above situation.

Property 6 :

Let l₁ and l₂ be two parallel lines and the line m intersects the lines l₁ and l₂.

The figure shown below illustrates the above situation.

From the above figure, we can have the following important results.

Properties of Perpendicular Lines

Property 1 :

Let m₁ and m₂ be the slopes of two lines.

If, the two lines are perpendicular, then the product of their slopes is equal to -1.

m₁ x m₂ = -1

Property 2 :

Let us consider the general form of equation of a straight line ax + by + c  = 0.

If the two lines are perpendicular, then their general form of equations will differ as shown below.

Property 3 :

Let us consider the slope intercept form of equation of a straight line y = mx + b.

If the two lines are perpendicular, then their slope-intercept form equations will differ as shown below.

Property 4 :

If the two lines are perpendicular, the angle between them will be 90°.

The figure shown below illustrates the above property.

Solving Problems Using Properties of Parallel and Perpendicular Lines

Problem 1 :

The slopes of the two lines are 7 and (3k + 2). If the two lines are parallel, find the value of k.

Solution :

If two lines are parallel, then their slopes are equal.

3k + 2 = 7

Subtract 2 from both sides.

3k = 5

Divide both sides by 5.

k = 5/3

Problem 2 :

If the following equations of two lines are parallel, then find the value of k.

3x + 2y - 8 = 0

(5k + 3)x + 2y + 1 = 0

Solution :

If the two lines are parallel, then their general forms of equations will differ only in the constant term and they will have the same coefficients of x and y.

To find the value of k, equate the coefficients of x.

5k + 3 = 3

Subtract 3 from both sides.

5k  =  0

Divide both sides by 5.

k = 0

Problem 3 :

The slopes of the two lines are 7 and (3k + 2). If the two lines are perpendicular, find the value of k.

Solution :

If the given two lines are perpendicular, then the product of the slopes is equal to -1.

7(3k + 2) = -1

Use distributive property.

21k + 14 = -1

Subtract 14 from each side.

21k = -15

Divide each side by 21.

k = -15/21

k = -5/7

Problem 4 :

The equations of the two perpendicular lines are

3x + 2y - 8 = 0

(5k + 3) - 3y + 1 = 0

Find the value of k.

Solution :

If the two lines are perpendicular, then the coefficient y term in the first line is equal to the coefficient of x term in the second line.

5k + 3 = 2

Subtract 3 from both sides.

5k = -1

Divide both sides by 5.

k = -1/5

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