Distance on the Coordinate Plane Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Distance on the Coordinate Plane.

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

Concept Recap

The distance between two points on the coordinate plane is found using the Pythagorean theorem: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ .

Draw a right triangle between the two points — the horizontal and vertical distances are the legs, and the straight-line distance is the hypotenuse.

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: Coordinate distance is Pythagorean theorem applied to horizontal and vertical changes.

Common stuck point: The procedure for distance on the coordinate plane is the easy part; the trap is adding run and rise. Asking "Can I draw horizontal and vertical legs between the points?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Can I draw horizontal and vertical legs between the points?

Worked Examples

Example 1

easy

Find the distance between

(1,2)

and

(1,-6)

. Show the shortcut for vertical segments.

Answer

First step

Notice both points have

x=1

— the segment is vertical.

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Example 2

medium

Express the distance between

(1,2)

and

(4,4)

in simplest radical form.

Example 3

medium

Three points:

A(0,0),\ B(4,0),\ C(4,3)

. Is triangle

ABC

a right triangle?

Example 4

medium

Find the distance between

(2,-1)

and

(7,11)

in simplest form.

Example 5

hard

Show that

(1,2),(4,6),(8,3)

form a right triangle.

Example 6

hard

A circle has center

(2,-1)

and radius 5. Determine whether

(5,3)

is inside, on, or outside the circle.

Example 7

challenge

Find the point on the

y

-axis equidistant from

(3,4)

and

(5,-2)

Practice Problems

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

Example 1

easy

Find the distance between

(1,1)

and

(4,5)

Example 2

easy

Find the distance between

(-2,-3)

and

(-2,4)

Example 3

easy

Find the distance between

(-4,2)

and

(3,2)

Example 4

easy

Find the distance between

(0,0)

and

(9,12)

Example 5

easy

Find the distance between

(-1,-1)

and

(2,3)

Example 6

medium

Find the distance between

(-3,2)

and

(5,-4)

Example 7

medium

A triangle has vertices

(0,0)

(6,0)

(0,8)

. Find its perimeter.

Example 8

medium

Find the length of the segment from

(-2,5)

(1,1)

Example 9

medium

Find the distance between

(-5,2)

and

(3,-4)

Example 10

medium

A square has vertices

(0,0),(3,0),(3,3),(0,3)

. Find its diagonal length.

Example 11

medium

A robot at

(2,3)

goes straight to a charging dock at

(14,-2)

. How far does it travel?

Example 12

hard

Find

y

so that

(3,y)

is exactly 5 units from

(0,1)

Example 13

hard

Quadrilateral has vertices

(0,0),(4,0),(4,3),(0,3)

. Find the length of each diagonal.

Example 14

hard

Is the triangle with vertices

(1,1),(4,5),(7,1)

isosceles, scalene, or equilateral?

Example 15

hard

Find the perimeter of triangle with vertices

(-1,-1),(2,3),(5,-1)

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Related Concepts

Coordinate Plane Pythagorean Theorem Square Roots Distance Formula Midpoint Formula

Background Knowledge

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

coordinate planepythagorean theoremsquare roots