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: Coordinates turn geometry into algebra—shapes become equations.

Common stuck point: The same shape has different equations in different coordinate systems.

Sense of Study hint: Try plotting a few points that satisfy the equation and connect the dots. The shape that appears is the geometric object.

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 to B.

Solution

  1. 1
    Step 1: Plot each point: A is 2 right, 5 up; B is 3 left, 1 up; C is on the y-axis, 4 down.
  2. 2
    Step 2: Distance AB = \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2} = \sqrt{(-3-2)^2 + (1-5)^2}.
  3. 3
    Step 3: = \sqrt{(-5)^2 + (-4)^2} = \sqrt{25 + 16} = \sqrt{41} \approx 6.40.

Answer

AB = \sqrt{41} \approx 6.40 units.
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.

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).

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.
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Background Knowledge

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

coordinate plane