Perpendicular lines intersect at 90° or right angle.

In the coordinate plane, they would look like as shown below.

If we take a closer look at these two lines, we see that the slope of one is 2 and the other is -1/2.

This can be generalized to any pair of perpendicular lines in the coordinate plane. The slopes of perpendicular lines are opposite signs and reciprocals of each other.

or

The product of slopes of any two perpendicular lines is always equal to -1.

In the above example, we have

(-1/2) x 2 = -1

**Postulate (Slopes of Perpendicular Lines) : **

In a coordinate plane, two lines are perpendicular if and only if the product of their slopes is -1.

The slope of a non vertical line is the ratio of the vertical change (the rise) to the horizontal change (the run).

If the line passes through the points (x_{1}, y_{1}) and (x_{2}, y_{2}), then the slope is given by

Slope = **rise / run**

**Slope = (y_{2} - y_{1}) / (x_{2} - x_{1})**

Usually, slope is represented by the variable m.

**Example 1 :**

In the diagram given below, find the slope of each line. Determine whether the lines j_{1 }and j_{2 }are perpendicular.

**Solution : **

**Part 1 :**

Find the slope of the line j_{1}. Line j_{1} is passing through the points (0, 3) and (3, 1).

Let (x_{1}, y_{1}) = (0, 3) and (x_{2}, y_{2}) = (3, 1)

Slope (j_{1}) = (y_{2} - y_{1}) / (x_{2} - x_{1})

Slope (j_{1}) = (1 - 3) / (3 - 0)

Slope (j_{1}) = - 2 / 3

**Part 2 :**

Find the slope of the line j_{2}. Line j_{2} is passing through the points (0, 3) and (-4, -3).

Let (x_{1}, y_{1}) = (0, 3) and (x_{2}, y_{2}) = (-4, -3)

Slope (j_{2}) = (-3 - 3) / (-4 - 0)

Slope (j_{2}) = (-6) / (-4)

Slope (j_{2}) = 3 / 2

**Multiply the slopes : **

The product is

= (-2/3) x (3/2)

= - 1

Since the product of slopes of the lines j_{1 and }j_{2 }is -1, the lines j_{1 and }j_{2} are perpendicular.

**Example 2 :**

In the diagram given below, find the slope of each line. Determine whether the lines are perpendicular.

**Solution : **

**Part 1 :**

Find the slope of the line AC. Line AC is passing through the points (1, -4) and (4, 2).

Let (x_{1}, y_{1}) = (1, -4) and (x_{2}, y_{2}) = (4, 2)

Slope (AC) = (y_{2} - y_{1}) / (x_{2} - x_{1})

Slope (AC) = [(2 - (-4)] / (4 - 1)

Slope (AC) = (2 + 4) / 3

Slope (AC) = 6 / 3

Slope (AC) = 2

**Part 2 :**

Find the slope of the line BD. Line BD is passing through the points (-1, 2) and (5, -1).

Let (x_{1}, y_{1}) = (-1, 2) and (x_{2}, y_{2}) = (5, -1)

Slope (BD) = (-1 - 2) / [(5 - (-1)]

Slope (BD) = (-3) / 6

Slope (BD) = -1 / 2

**Multiply the slopes : **

The product is

= (2) x (-1/2)

= - 1

Since the product of slopes of the lines is -1, the lines AC and BD are perpendicular.

**Example 3 :**

Decide whether the lines are perpendicular.

Line 1 : y = 3x/4 + 2

Line 2 : y = -4x/3 - 3

**Solution : **

When we compare the given equations to slope intercept equation of a line y = mx + b, we get

slope of line 1 = 3/4

slope of line 2 = -4/3

Multiply the slopes :

The product is

= (3/4) x (-4/3)

= - 1

Since the product of slopes of the lines is -1, the given lines are perpendicular.

**Example 4 :**

Decide whether the lines are perpendicular.

Line 1 : 4x + 5y = 2

Line 2 : 5x + 4y = 3

**Solution : **

Rewrite each equation in slope-intercept form to find the slope.

4x + 5y = 2 5y = -4x + 2 y = -4x/5 + 2/5 Slope = -4/5 |
5x + 4y = 3 4y = -5x + 3 y = -5x/4 + 3/4 Slope = -5/4 |

**Multiply the slopes : **

The product is

= (-4/5) x (-5/4)

= 1

Since the product of slopes of the lines is not -1, the given lines are not perpendicular.

**Example 5 :**

In the diagram given below, the equation y = 3x/2 + 3 represents a mirror. A ray of light hits the mirror at (-2, 0). What is the equation of the line p that is perpendicular to the mirror at this point ?

**Solution : **

The slope of the mirror is 3/2. So, the slope of the line p is -2/3.

Let y = mx + b be the equation of the line p.

Substitute (x, y) = (-2, 0) and m = -2/3 to find the value of b.

0 = (-2/3)(-2) + b

0 = 4/3 + b

Subtract 4/3 from both sides.

-4/3 = b

So, the equation of the line p is

y = -2x/3 - 4/3

Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.

HTML Comment Box is loading comments...

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 **

**Word problems on quadratic equations**

**Area and perimeter word problems**

**Word problems on direct variation and inverse variation **

**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**

**Trigonometry word problems**

**Markup and markdown word problems **

**Word problems on mixed fractrions**

**One step equation word problems**

**Linear inequalities word problems**

**Ratio and proportion word problems**

**Word problems on sets and venn diagrams**

**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 **

**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 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**