Perpendicularity Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Perpendicularity.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Lines, segments, or planes that intersect at exactly a right angle of $90°$ to each other.

The corner of a book or a room—the two edges meet at precisely $90°$ .

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Two lines are perpendicular when they cross at exactly $90^\circ$ — slopes that multiply to $-1$ .

Common stuck point: The procedure for perpendicularity is the easy part; the trap is using equal slopes as the test. Asking "Do the two lines meet at exactly $90^\circ$ , with slopes multiplying to $-1$ ?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Do the two lines meet at exactly $90^\circ$ , with slopes multiplying to $-1$ ?

Worked Examples

Example 1

easy

Line

\ell_1: y = 2x + 1

. Write the equation of a line

\ell_2

perpendicular to

\ell_1

that passes through

(4, 3)

Answer

y = -\dfrac{1}{2}x + 5

First step

Step 1: Slope of

\ell_1

m_1 = 2

Full solution

2
Step 2: Perpendicular slope: $m_2 = -\dfrac{1}{m_1} = -\dfrac{1}{2}$ (since $m_1 \times m_2 = -1$ ).
3
Step 3: Point-slope form: $y - 3 = -\dfrac{1}{2}(x - 4) \Rightarrow y = -\dfrac{1}{2}x + 5$ .

Perpendicular lines meet at

90°

. Their slopes are negative reciprocals: if one slope is

m

, the other is

-1/m

. This ensures

m_1 \times m_2 = -1

Example 2

medium

Determine whether triangle

A(0,0)

B(4,0)

C(4,3)

is a right triangle, and if so identify the right angle vertex.

Triangle A(0,0), B(4,0), C(4,3) — identify the right angle

Example 3

medium

Write the equation of the line through

(1, 2)

perpendicular to

y = 3x + 4

Example 4

medium

Write the equation of the line through

(4, 6)

perpendicular to

y = -2x + 1

Example 5

hard

Find the foot of the perpendicular from

(5, 3)

to the line

y = x

Example 6

hard

Find the foot of the perpendicular from the origin to the line

3x + 4y = 25

Example 7

challenge

A triangle has vertices

A(1, 1)

B(7, 3)

C(5, 9)

. Find the equation of the altitude from

C

AB

Example 8

challenge

Given line

\ell: 4x + 3y = 12

, find the equation of the line perpendicular to

\ell

through the point where

\ell

crosses the

x

-axis.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Line

m

passes through

(0, -1)

with slope

\dfrac{3}{5}

. Write the equation of the line perpendicular to

m

through the origin.

Example 2

hard

Find the foot of the perpendicular from point

P(3, 7)

to the line

\ell: y = 2x - 1

Example 3

easy

At what angle do perpendicular lines meet?

Example 4

easy

Line A has slope 2. What slope makes a line perpendicular to A?

Example 5

easy

Two lines have slopes 3 and

-\tfrac{1}{3}

. Are they perpendicular?

Example 6

easy

Is a horizontal line perpendicular to a vertical line?

Example 7

easy

What is the product of the slopes of two perpendicular (non-vertical) lines?

Example 8

easy

Line A has slope

-\tfrac{2}{3}

. Find the slope perpendicular to A.

Example 9

easy

A square has perpendicular sides. How many right angles does it have?

Example 10

easy

Are lines with slopes 2 and

-2

perpendicular?

Example 11

medium

Write the equation of the line through

(0, 2)

perpendicular to

y = \tfrac{1}{4}x + 5

Example 12

medium

Find the slope perpendicular to the line through

(2, 1)

and

(6, 9)

Example 13

medium

Why are perpendicular slopes negative reciprocals rather than just opposites?

Example 14

medium

Is the line

y = 5x

perpendicular to

5y = -x

Example 15

medium

The diagonals of a rhombus are perpendicular. If one diagonal is horizontal, what is true of the other?

Example 16

medium

A perpendicular bisector of a segment does what to the segment?

Example 17

medium

A right triangle's legs are perpendicular. If one leg lies along a line of slope 4, what is the slope of the other leg?

Example 18

medium

Use slopes to check if the triangle with vertices

A(0,0)

B(4,0)

C(4,3)

has a right angle at

B

Triangle A(0,0), B(4,0), C(4,3) — verify the right angle at B

Example 19

challenge

Find the equation of the perpendicular bisector of the segment from

(2, 1)

(6, 5)

Example 20

challenge

Find the foot of the perpendicular from the point

(0, 0)

to the line

y = x + 4

Example 21

challenge

Show that a triangle with vertices

(1,1)

(5,1)

(1,4)

is a right triangle using the slope test.

Triangle with vertices (1,1), (5,1), (1,4) — identify the right angle vertex

Example 22

challenge

Why does the perpendicular distance give the SHORTEST distance from a point to a line?

Example 23

easy

Find the slope of a line perpendicular to one with slope

5

Example 24

easy

Find the slope perpendicular to a line with slope

-\tfrac{3}{4}

Example 25

easy

Are the lines

y = 6x + 1

and

y = 6x - 5

perpendicular?

Example 26

easy

Find a slope perpendicular to one with slope

\tfrac{7}{2}

Example 27

medium

Lines

\ell_1

and

\ell_2

pass through

(0,0)

and

(2,8)

, and

(0,5)

and

(4,4)

respectively. Are they perpendicular?

Example 28

medium

Find the equation of the perpendicular bisector of the segment from

(1, 3)

(5, 7)

Example 29

medium

In the triangle with vertices

A(1, 2)

B(4, 2)

C(4, 8)

, identify the right-angle vertex using slopes.

Example 30

medium

Line

2x - 5y = 10

has what slope? What slope is perpendicular?

Example 31

medium

In a

30°-60°-90°

triangle, are the two legs perpendicular?

Example 32

hard

Find the distance from the point

(0, 0)

to the line

2x - y + 10 = 0

Example 33

hard

Show that the triangle with vertices

(0,0)

(3,4)

(-4,3)

has a right angle at the origin.

Example 34

hard

Line

\ell

has equation

y = \tfrac{2}{3}x - 1

. Write the equation of the line perpendicular to

\ell

that passes through the

y

-intercept of

\ell

Example 35

hard

In the quadrilateral with vertices

A(0,0)

B(4,0)

C(4,3)

D(0,3)

, are sides

AB

and

BC

perpendicular?

Example 36

challenge

Two lines are perpendicular and pass through

(2, 5)

. One has slope

\tfrac{2}{3}

. Find the

x

-intercept of the other.

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Related Concepts

Line Slope Angles Perpendicularity

Background Knowledge

These ideas may be useful before you work through the harder examples.

lineslopeangles