# SLOPES OF PARALLEL AND PERPENDICULAR LINES

Slopes of Parallel Lines :

If two non vertical straight lines are parallel, then slopes of the lines will be equal.

If two non vertical straight lines with slopes m1 and m2, are parallel, then

m1  =  m2

On the other hand, if the slopes of two straight lines are equal, then the lines will be parallel.

Slopes of Perpendicular Lines :

If two non vertical straight lines are perpendicular, then the product of slopes of the lines will be equal to '-1'.

If two non vertical straight lines with slopes m1 and m2, are perpendicular, then

m1 m2  =  –1

On the other hand, if the product of slopes of two straight lines is equal to '-1', then the lines will be perpendicular.

Note :

The straight lines x-axis and y-axis are perpendicular to each other. But, the condition m1m2 = –1 is not true. Because the slope of the x-axis is zero and the slope of the y-axis is not defined.

## Examples

Example 1 :

The line joining the points A (-2 , 3) and B (a , 5) is parallel to the line joining the points C (0 , 5) and D (-2 , 1). Find the value of a.

Solution :

Since the line joining the points AB and CD are parallel, the slope of those two lines will be equal.

Slope of AB :

A(-2 , 3) ==> (x1, y1),  B(a , 5)  ==>  (x2, y2)

Slope (m1)  =  (y2 - y1)/(x2 - x1)

=  (5 - 3) / (a - (-2))

=  2 / (a + 2)

Slope of CD :

C(0 , 5) ==> (x1, y1),  D(-2 , 1) ==>  (x2, y2)

Slope (m2)  =  (y2 - y1)/(x2 - x1)

=  (1 - 5) / (-2 - 0)

=  -4 / (-2)

=  2

Slope of AB  =  Slope of CD

2 / (a + 2)  =  2

Multiply by (a +2) on both sides

2  =  2(a + 2)

2  =  2a + 4

Subtract 4 on both sides

2 - 4  =  2a

2a  =  -2

Divide by 2 on both sides,

a  =  -1

Example 2 :

The line joining the points A(0, 5) and B (4, 2) is perpendicular to the line joining the points C (-1, -2) and D (5, b). Find the value of b.

Solution :

Since the line joining the points AB and CD are perpendicular, the product of slopes will be equal to -1.

Slope of AB :

A (0, 5) ==> (x1, y1),  B (4, 2) ==>  (x2, y2)

Slope (m1)  =  (y2 - y1)/(x2 - x1)

=  (2 - 5) / (4 - 0)

=  -3 / 4

Slope of CD :

C (-1, -2) ==> (x, y),  D (5 , b) ==>  (x, y)

Slope (m2)  =  (y2 - y1)/(x2 - x1)

=  (b - (-2)) / (5 - (-1))

=  (b + 2) / (5 + 1)

=  (b + 2) / 6

Slope of AB x Slope of CD  =  -1

(-3/4)  x  (b + 2)/6  =  -1

(b + 2)/8  =  1

Multiply by 8 on both sides,

b + 2  =  8

Subtract by 2 on both sides

b  =  8 - 2

b  =  6

Example 3 :

The vertices of triangle ABC are A(1, 8), B(-2, 4), C(8, -5). If M and N are the midpoints of AB and AC respectively, find the slope of MN and hence verify that MN is parallel to BC.

Solution :

Mid point of the side AB  =  M

mid point  =  (x1 + x2)/2 ,  (y1 + y2)/2

=  [1 + (-2)]/2 , (8 + 4)/2

=  (-1/2, 6)

Mid point of the side AC  =  N

mid point  =  (x1 + x2)/2 ,  (y1 + y2)/2

=  [1 + 8]/2 , (8 - 5)/2

=  (9/2, 3/2)

Slope of BC :

=   (y2 - y1)/(x2 - x1)

=  (-5 -4) / (8 + 2)

=  -9 / 10  ------(1)

Slope of MN :

=   (y2 - y1)/(x2 - x1)

= ((3/2) - 6) / ((9/2) + (1/2))

= (-9/2) /  (10/2)

=  -9 / 10  ------(2)

So, the sides MN and BC are parallel. Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.

If you have any feedback about our math content, please mail us :

v4formath@gmail.com

You can also visit the following web pages on different stuff in math.

WORD PROBLEMS

Word problems on simple equations

Word problems on linear equations

Algebra word problems

Word problems on trains

Area and perimeter word problems

Word problems on direct variation and inverse variation

Word problems on unit price

Word problems on unit rate

Word problems on comparing rates

Converting customary units word problems

Converting metric units word problems

Word problems on simple interest

Word problems on compound interest

Word problems on types of angles

Complementary and supplementary angles word problems

Double facts word problems

Trigonometry word problems

Percentage word problems

Profit and loss word problems

Markup and markdown word problems

Decimal word problems

Word problems on fractions

Word problems on mixed fractrions

One step equation word problems

Linear inequalities word problems

Ratio and proportion word problems

Time and work word problems

Word problems on sets and venn diagrams

Word problems on ages

Pythagorean theorem word problems

Percent of a number word problems

Word problems on constant speed

Word problems on average speed

Word problems on sum of the angles of a triangle is 180 degree

OTHER TOPICS

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

Time, speed and distance shortcuts

Ratio and proportion shortcuts

Domain and range of rational functions

Domain and range of rational functions with holes

Graphing rational functions

Graphing rational functions with holes

Converting repeating decimals in to fractions

Decimal representation of rational numbers

Finding square root using long division

L.C.M method to solve time and work problems

Translating the word problems in to algebraic expressions

Remainder when 2 power 256 is divided by 17

Remainder when 17 power 23 is divided by 16

Sum of all three digit numbers divisible by 6

Sum of all three digit numbers divisible by 7

Sum of all three digit numbers divisible by 8

Sum of all three digit numbers formed using 1, 3, 4

Sum of all three four digit numbers formed with non zero digits

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6

Featured Categories

Math Word Problems

SAT Math Worksheet

P-SAT Preparation

Math Calculators

Quantitative Aptitude

Transformations

Algebraic Identities

Trig. Identities

SOHCAHTOA

Multiplication Tricks

PEMDAS Rule

Types of Angles

Aptitude Test 