Analytic Geometry Examples in Math

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

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

Concept Recap

Analytic geometry studies geometric objects using coordinate systems and algebraic equations.

It translates shapes into equations so algebra can solve geometry problems.

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 geometric figures into algebraic equations, making them solvable with algebra tools.

Common stuck point: Students lose geometric meaning when manipulating equations.

Sense of Study hint: Interpret each algebra step on the graph to keep the geometry visible.

Worked Examples

Example 1

medium
Prove that the triangle with vertices A(0,0), B(4,0), and C(2,2\sqrt{3}) is equilateral using coordinates.

Solution

  1. 1
    Compute AB: AB = \sqrt{(4-0)^2 + (0-0)^2} = \sqrt{16} = 4.
  2. 2
    Compute BC: BC = \sqrt{(2-4)^2 + (2\sqrt{3}-0)^2} = \sqrt{4 + 12} = \sqrt{16} = 4.
  3. 3
    Compute CA: CA = \sqrt{(0-2)^2 + (0-2\sqrt{3})^2} = \sqrt{4+12} = 4.
  4. 4
    Since AB = BC = CA = 4, all three sides are equal, so the triangle is equilateral.

Answer

The triangle is equilateral with all sides equal to 4.
Analytic geometry lets us prove geometric properties by computing distances with the distance formula. Equal side lengths confirm an equilateral triangle without any angle measurements.

Example 2

hard
Use coordinates to prove that the diagonals of a rectangle bisect each other.

Practice Problems

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

Example 1

easy
Determine whether points P(1,2), Q(3,6), and R(5,10) are collinear using the slope method.

Example 2

medium
Show that quadrilateral A(1,1), B(4,2), C(5,5), D(2,4) is a parallelogram by comparing slopes of opposite sides.
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Background Knowledge

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

coordinate representationdistance formulaslope in geometry