Coordinate Plane Examples in Math

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

A two-dimensional surface formed by two perpendicular number lines — the horizontal $x$ -axis and the vertical $y$ -axis — intersecting at the origin $(0, 0)$ . Every point on the plane is uniquely identified by an ordered pair $(x, y)$ giving its horizontal and vertical distances from the origin.

Like a map with street numbers—the address $(3, 2)$ is 3 right, 2 up.

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: The coordinate plane locates every point by an ordered pair $(x,y)$ measured from the origin.

Common stuck point: The procedure for coordinate plane is the easy part; the trap is reversing the coordinates. Asking "Am I locating or drawing a position using a horizontal value paired with a vertical value?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I locating or drawing a position using a horizontal value paired with a vertical value?

Worked Examples

Example 1

easy

In which quadrant is the point

(-3, 5)

located?

The point (−3, 5) is in Quadrant II: negative x, positive y.

Answer

Quadrant II

First step

The

x

-coordinate is

-3

(negative, so left of the origin).

Full solution

2
The $y$ -coordinate is $5$ (positive, so above the origin).
3
Negative $x$ and positive $y$ place the point in Quadrant II.

The coordinate plane has four quadrants. The signs of the coordinates determine the quadrant: (+,+) is QI, (−,+) is QII, (−,−) is QIII, and (+,−) is QIV.

Example 2

medium

Find the distance between points

(1, 2)

and

(4, 6)

Distance from A(1, 2) to B(4, 6) using the distance formula.

Example 3

easy

Name the point that is

5

units directly below the origin.

Example 4

medium

A point starts at

(2, -1)

, moves

3

units right and

4

units up. Where does it end?

Example 5

medium

Which is closer to the origin:

(6, 8)

(7, 7)

Example 6

medium

A rectangle has corners

(0,0)

(5,0)

(5,3)

(0,3)

. What is the length of its diagonal?

Diagonal d = √(5² + 3²) = √34.

Example 7

hard

M(4, 1)

is the midpoint of segment

AB

with

A(-2, 5)

, find

B

Practice Problems

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

Example 1

easy

Plot the point

(2, -3)

. Is it above or below the

x

-axis?

The point (2, −3) lies below the x-axis (Quadrant IV).

Example 2

medium

Find the midpoint of the segment connecting

(-2, 4)

and

(6, 0)

Midpoint M(2, 2) of segment AB.

Example 3

easy

What are the coordinates of the origin?

Example 4

easy

In the point

(3, 5)

, which number is the

x

-coordinate?

Example 5

easy

Which quadrant contains

(-2, 3)

Sign pattern (−, +) places (−2, 3) in Quadrant II.

Example 6

easy

Plot direction: to reach

(0, -4)

from the origin, move which way?

From the origin, move down 4 to reach (0, −4).

Example 7

easy

On which axis does

(7, 0)

lie?

y = 0 means (7, 0) lies on the x-axis.

Example 8

easy

Are

(2, 3)

and

(3, 2)

the same point?

Example 9

easy

How far above the

x

-axis is the point

(5, 4)

The y-coordinate 4 is the height of (5, 4) above the x-axis.

Example 10

easy

What is the sign pattern of coordinates in Quadrant III?

Example 11

medium

Find the distance between

(1, 2)

and

(4, 6)

Distance = √(3² + 4²) = 5.

Example 12

medium

Find the midpoint of

(2, 4)

and

(8, 10)

Midpoint M(5, 7) is the average of A(2,4) and B(8,10).

Example 13

medium

A point is

3

left and

2

down from

(5, 5)

. Find it.

Example 14

medium

Points

(2,1)

and

(2,7)

— what kind of segment connects them?

Same x-coordinate (x = 2) means the segment is vertical.

Example 15

medium

Which point is farther from the origin:

(3, 4)

(0, 6)

B(0,6) is farther: distance 6 vs A's distance 5.

Example 16

medium

If a point in Quadrant I is reflected over the

x

-axis, which quadrant results?

Reflection over x-axis: (x, y) → (x, −y). Quadrant I → Quadrant IV.

Example 17

medium

Plot

(x, y)

where

y = 2x

and

x = 3

. Where is it?

Example 18

challenge

A triangle has vertices

(0,0)

(4,0)

(4,3)

. Find its area.

Right triangle with base 4 and height 3; area = ½ × 4 × 3 = 6.

Example 19

challenge

Find the point that divides the segment from

(1,2)

(7,8)

in ratio

1:2

from the first point.

Example 20

challenge

Three points

(1,1)

(2,3)

(3,5)

— are they collinear?

Equal slopes between consecutive points confirm collinearity.

Example 21

medium

Reflect

(4, -3)

over the

y

-axis.

Reflection over y-axis: (4, −3) → (−4, −3).

Example 22

medium

What is the perimeter path length from

(0,0)

(3,0)

(3,4)

Path length = 3 + 4 = 7.

Example 23

easy

In which quadrant is the point

(5, -2)

Sign pattern (+, −) places (5, −2) in Quadrant IV.

Example 24

easy

Starting at the origin, move right

4

and up

6

. Where do you land?

Example 25

easy

Is the point

(-6, 0)

on the

x

-axis or the

y

-axis?

Example 26

easy

How far to the left of the

y

-axis is

(-7, 3)

Example 27

medium

Find the distance between

(-2, 1)

and

(3, 13)

Distance = √(5² + 12²) = √169 = 13.

Example 28

medium

Find the midpoint of the segment from

(-5, 3)

(7, -9)

Midpoint M(1, −3) of segment from A(−5, 3) to B(7, −9).

Example 29

medium

Reflect

(-3, 5)

over the line

y = x

Example 30

medium

Two points are

(1, 4)

and

(1, -2)

. What is the length of the segment between them?

Vertical segment at x = 1 has length |4 − (−2)| = 6.

Example 31

medium

Point

(a, b)

is in Quadrant III. In which quadrant is

(-a, b)

Example 32

medium

Find the midpoint of the segment from

(-4, -6)

(10, 2)

Example 33

medium

Are the points

(2, 3)

(4, 3)

(4, 7)

vertices of a right triangle?

Horizontal leg PQ and vertical leg QR meet at a right angle at Q.

Example 34

hard

Find the perimeter of the triangle with vertices

(0,0)

(6,0)

(0,8)

Perimeter = 6 + 8 + 10 = 24.

Example 35

hard

A square has opposite vertices

(1, 2)

and

(5, 6)

. Find the length of its diagonal.

Diagonal d = √(4² + 4²) = 4√2.

Example 36

hard

Find the area of the triangle with vertices

(1, 1)

(5, 1)

(3, 7)

Area = ½ × base × height = ½ × 4 × 6 = 12.

Example 37

challenge

Find the point on the

x

-axis equidistant from

(1, 2)

and

(5, 4)

Example 38

challenge

Find the area of the quadrilateral with vertices

(0,0)

(4,0)

(5,3)

(1,3)

Parallelogram with base 4 and height 3; area = 4 × 3 = 12.

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Related Concepts

Number Sense Integers Distance Formula Slope

Background Knowledge

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

number senseintegers