Parallel and Perpendicular Examples in Math

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

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

Concept Recap

Parallel lines never intersect and have matching direction; perpendicular lines intersect at right angles.

Parallel tracks run side by side; perpendicular streets form a plus sign.

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: Parallel lines never meet (equal slopes); perpendicular lines cross at right angles (slopes multiply to $-1$ ).

Common stuck point: The procedure for parallel and perpendicular is the easy part; the trap is thinking same-sign slopes like $2$ and $-2$ are perpendicular. Asking "Are the two lines' slopes equal (parallel) or negative reciprocals so their product is $-1$ (perpendicular)?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Are the two lines' slopes equal (parallel) or negative reciprocals so their product is $-1$ (perpendicular)?

Worked Examples

Example 1

easy

Find the equation of the line through

(3, -1)

that is parallel to

y = 2x + 5

The new line through (3, −1) is parallel to y = 2x + 5.

Answer

y = 2x - 7

First step

Parallel lines have equal slopes. The slope of

y = 2x + 5

m = 2

, so the new line also has

m = 2

Full solution

2
Use point-slope form: $y - (-1) = 2(x - 3)$ .
3
Simplify: $y + 1 = 2x - 6$ , so $y = 2x - 7$ .

Parallel lines never intersect because they have identical slopes. To find a parallel line through a specific point, keep the slope the same and use point-slope form to determine the new

y

-intercept.

Example 2

medium

Find the equation of the line through

(4, 1)

that is perpendicular to

3x - y = 6

The new line through (4, 1) is perpendicular to 3x − y = 6.

Example 3

medium

Find the equation of the line through

(2,-7)

parallel to

y=-3x+1

Find the line through (2, −7) parallel to y = −3x + 1.

Example 4

medium

Find the equation of the line through

(-3,2)

perpendicular to

y=\tfrac{1}{4}x-7

Example 5

medium

Show that the line through

(1,3)

and

(5,11)

is parallel to the line through

(0,-2)

and

(2,2)

Example 6

medium

Determine whether the quadrilateral with vertices

A(0,0)

B(4,0)

C(5,3)

D(1,3)

is a parallelogram.

Is ABCD a parallelogram? Verify using slopes.

Example 7

hard

Determine whether

A(0,0)

B(4,2)

C(3,4)

form a right triangle.

Does triangle ABC have a right angle? If so, at which vertex?

Example 8

hard

Find the equation of the perpendicular bisector of the segment from

(2,1)

(8,5)

Example 9

hard

Show that the quadrilateral with vertices

A(1,0)

B(0,2)

C(2,3)

D(3,1)

is a rectangle.

Example 10

medium

Find the distance from the point

(0,0)

to the line

y=2x+5

using the perpendicular line.

Example 11

challenge

Show that the points

A(1,2)

B(4,3)

C(5,6)

D(2,5)

form a rhombus (all sides equal) but not a square (no right angle).

Practice Problems

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

Example 1

easy

Are the lines

y = -4x + 1

and

y = \frac{1}{4}x - 3

parallel, perpendicular, or neither?

Are these lines parallel, perpendicular, or neither?

Example 2

medium

Line

\ell

passes through

(0,3)

and

(6,0)

. Find the equation of the line perpendicular to

\ell

passing through the origin.

The line through the origin perpendicular to ℓ has equation y = 2x.

Example 3

easy

Are

y=\tfrac{1}{2}x+4

and

y=\tfrac{1}{2}x-7

parallel, perpendicular, or neither?

Example 4

easy

Are

y=4x-1

and

y=-\tfrac{1}{4}x+9

parallel, perpendicular, or neither?

Example 5

easy

Are

y=2x+3

and

y=3x+2

parallel, perpendicular, or neither?

Example 6

medium

Find the equation of the line through

(0,-2)

parallel to

2x+5y=10

Example 7

medium

Find the equation of the line through

(1,4)

perpendicular to

x-2y=8

Find the line through (1, 4) perpendicular to x − 2y = 8.

Example 8

medium

Are the lines through

(0,0)

(3,2)

and through

(1,5)

(-1,8)

parallel, perpendicular, or neither?

Are these two line directions parallel, perpendicular, or neither?

Example 9

hard

Find

k

so that lines

y=kx+1

and

y=(k-2)x-3

are perpendicular.

Example 10

hard

Find

a

such that the line

ax+3y=6

is parallel to the line

4x-y=7

Example 11

medium

Find the equation of the line through

(-2,3)

that is perpendicular to the

x

-axis.

Example 12

medium

Find the equation of the line through

(5,-4)

parallel to the

x

-axis.

Example 13

hard

The line

\ell

passes through

(2,1)

and

(6,4)

. Find the equation of the line through

(0,0)

perpendicular to

\ell

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Related Concepts

Angles Line Slope in Geometry

Background Knowledge

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

angleslineslope in geometry