Coordinate Representation Examples in Math

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

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

Concept Recap

Describing geometric objects precisely using ordered pairs $(x, y)$ or triples $(x, y, z)$ in a coordinate system.

Every point has a unique numerical 'address' like $(3, 4)$ that locates it exactly on the plane.

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 representation pins each geometric point to an ordered pair $(x,y)$ or triple $(x,y,z)$ so geometry becomes algebra.

Common stuck point: The procedure for coordinate representation is the easy part; the trap is swapping $x$ and $y$ . Asking "Am I assigning exact numeric coordinates to points so geometry can be done with algebra?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I assigning exact numeric coordinates to points so geometry can be done with algebra?

Worked Examples

Example 1

easy

Plot points

A(2, 5)

B(-3, 1)

C(0, -4)

on the coordinate plane and find the distance from

A

B

Plot of A, B, C with distance AB = √41

Answer

AB = \sqrt{41} \approx 6.40

units.

First step

Step 1: Plot each point:

A

2

right,

5

up;

B

3

left,

1

up;

C

is on the

y

-axis,

4

down.

Full solution

2
Step 2: Distance $AB = \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2} = \sqrt{(-3-2)^2 + (1-5)^2}$ .
3
Step 3: $= \sqrt{(-5)^2 + (-4)^2} = \sqrt{25 + 16} = \sqrt{41} \approx 6.40$ .

The coordinate plane assigns a unique ordered pair

(x, y)

to every point. The distance formula is an application of the Pythagorean theorem: the horizontal and vertical differences form the legs of a right triangle.

Example 2

medium

Write the equation of the circle with centre

(-2, 3)

and radius

5

in standard form. Verify that the point

(3, 3)

lies on the circle.

Circle center (−2, 3), radius 5; point (3, 3) lies on the circle

Example 3

medium

Translate the geometric statement 'this point is

4

units from

(1, 2)

' into a coordinate equation.

Example 4

medium

The point

(1, 2, -3)

lies in 3D space. What are its projections onto the three coordinate planes?

Example 5

hard

Find the equation of the perpendicular bisector of the segment from

(2, 1)

(8, 7)

Perpendicular bisector of AB passes through M(5,4) with slope −1

Example 6

hard

Show that the points

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

form a rectangle (not just a parallelogram).

Axis-aligned sides → all four angles are 90°

Practice Problems

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

Example 1

easy

Find the midpoint of the segment joining

P(-4, 6)

and

Q(8, -2)

Midpoint M = (2, 2) of segment PQ

Example 2

hard

Three vertices of a parallelogram are

A(1,2)

B(5,2)

C(7,6)

. Using coordinate methods, find the fourth vertex

D

and compute the area of the parallelogram.

Parallelogram ABCD: D = (3, 6), area = 16 sq units

Example 3

easy

In the point

(3, 5)

, which number is the

y

-coordinate?

Example 4

easy

Are

(3, 5)

and

(5, 3)

the same point?

Example 5

easy

What equation represents a circle of radius 5 centered at the origin?

Example 6

easy

What does the equation

y = 2x + 3

represent geometrically?

Example 7

easy

How many numbers locate a point in 3D space?

Example 8

easy

Which point is in the second quadrant:

(2, 3)

(-2, 3)

, or

(-2, -3)

(-2, 3) is in Quadrant II: negative x, positive y

Example 9

easy

What are the coordinates of the origin?

Example 10

easy

Does the point

(4, 0)

lie on the

x

-axis or the

y

-axis?

Example 11

medium

Translate the claim 'this point is 5 units from the origin' into a coordinate equation.

Example 12

medium

What does the equation

(x - 2)^2 + (y - 3)^2 = 16

represent?

Circle center (2, 3), radius 4

Example 13

medium

Find the midpoint of the segment from

(2, 4)

(8, 10)

Midpoint M = (5, 7) of segment AB

Example 14

medium

Why does coordinate geometry let you prove geometric facts with algebra?

Example 15

medium

The points

(1,1)

(4,1)

(4,5)

(1,5)

are given. What shape do they form?

The four points form a 3×4 rectangle

Example 16

medium

Write the equation of the vertical line through

(7, 2)

Example 17

medium

Does the point

(3, 4)

lie on the circle

x^2 + y^2 = 25

Point (3, 4) lies on the circle x² + y² = 25

Example 18

medium

A point in 3D is

(2, 5, 9)

. What is its projection onto the

xy

-plane, in coordinates?

Example 19

challenge

Prove that the triangle with vertices

(0,0)

(6,0)

(3, 3\sqrt{3})

is equilateral using coordinates.

All sides = 6; the triangle is equilateral

Example 20

challenge

The equation

x^2 + y^2 - 6x + 8 = 0

represents a circle. Find its center and radius by completing the square.

Example 21

challenge

Show that the points

(0,0)

(4,0)

(4,3)

(0,3)

form a rectangle, and find its area, using coordinates.

Rectangle with sides 4 and 3; area = 12

Example 22

challenge

Why does adding a

z

-coordinate let you represent objects that 2D coordinates cannot?

Example 23

easy

Write the equation of the circle centered at the origin with radius

7

Example 24

easy

Which point is in Quadrant IV:

(-1, -1)

(3, -5)

, or

(0, 4)

(3, −5) is in Quadrant IV: positive x, negative y

Example 25

easy

Does the point

(-3, 4)

lie on the circle

x^2 + y^2 = 25

(-3, 4) lies on x² + y² = 25 since 9 + 16 = 25

Example 26

easy

What does

y = -3x

represent geometrically?

Example 27

medium

Write the equation of the circle with center

(3, -4)

and radius

5

Circle center (3, −4), radius 5

Example 28

medium

Find the

y

-intercept of the line

3x + 2y = 12

y-intercept of 3x + 2y = 12 is (0, 6)

Example 29

medium

Does

(2, -3)

lie on the line

y = -2x + 1

Example 30

medium

What shape do the points

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

form?

Right triangle: legs 3 and 4, right angle at origin

Example 31

hard

Find center and radius of the circle

x^2 + y^2 - 4x + 6y - 3 = 0

Example 32

hard

Find the equation of the line through

(1, 2)

perpendicular to

y = \frac{1}{3}x + 5

Example 33

hard

Convert the polar point

(r, \theta) = (2, \pi/3)

to Cartesian coordinates.

Example 34

hard

Write the equation of the sphere with center

(0, 0, 0)

and radius

r

Example 35

hard

Find the area of the triangle with vertices

(0,0), (5,0), (2, 6)

Triangle area = ½ × 5 × 6 = 15

Example 36

challenge

Convert the Cartesian point

(-2, 2)

to polar coordinates with

r > 0

and

0 \le \theta < 2\pi

Example 37

challenge

Find the shortest distance from the point

(4, 5)

to the line

3x + 4y - 12 = 0

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Related Concepts

Coordinate Plane Analytic Geometry Parametric Equations

Background Knowledge

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

coordinate plane