Coordinate Proofs Examples in Math

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

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 method of proving geometric properties by placing figures on a coordinate plane and using algebraic formulas (distance, midpoint, slope) to verify relationships.

Instead of arguing with angles and congruence marks, drop the shape onto a grid and let algebra do the heavy lifting. Want to prove a quadrilateral is a parallelogram? Calculate all four slopes—if opposite sides have equal slopes, they're parallel, and you're done. Coordinates turn visual intuition into airtight calculation.

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 proofs verify geometric claims by placing the figure on axes and computing distances, slopes, and midpoints.

Common stuck point: The procedure for coordinate proofs is the easy part; the trap is using specific numbers to prove a general theorem. Asking "Am I proving a geometric property by assigning coordinates and computing distance, slope, or midpoint?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I proving a geometric property by assigning coordinates and computing distance, slope, or midpoint?

Worked Examples

Example 1

medium

Use a coordinate proof to show that the diagonals of a rectangle are equal in length. Place the rectangle with one vertex at the origin.

Answer

Both diagonals equal

\sqrt{a^2 + b^2}

, so they are congruent.

First step

Step 1: Assign coordinates strategically. Place the rectangle with vertices at

A(0,0)

B(a,0)

C(a,b)

, and

D(0,b)

, where

a,b > 0

See the full worked solution + why-it-works coaching

SetupKey insightWhy it worksCommon pitfallConnection

Unlock answer keys One Family plan — every worked solution, all subjects

Example 2

hard

Use a coordinate proof to show that the midpoints of the sides of any quadrilateral form a parallelogram (Varignon's Theorem).

Example 3

medium

Using coordinates

A(0,0), B(2a,0), C(2a,2b), D(0,2b)

, show

AB

and

CD

are congruent.

Example 4

hard

Prove using coordinates that an isosceles triangle with vertices

A(-a,0), B(a,0), C(0,h)

has the perpendicular from

C

bisecting

AB

Prove the altitude from C bisects AB by showing its foot equals the midpoint of AB.

Practice Problems

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

Example 1

medium

Use a coordinate proof to show that the segment connecting the midpoints of two sides of a triangle is parallel to and half the length of the third side (Midsegment Theorem). Use triangle with vertices

A(0,0)

B(2a,0)

C(2b,2c)

Example 2

easy

Prove using coordinates that the diagonals of a square are perpendicular. Place the square with vertices at

(0,0)

(a,0)

(a,a)

(0,a)

Example 3

easy

Two segments have slopes

\frac{3}{4}

and

-\frac{4}{3}

. Are they parallel, perpendicular, or neither?

Example 4

medium

Prove the triangle

(0,0)

(5,0)

(0,12)

is right-angled.

Prove the triangle is right-angled at the origin.

Example 5

medium

Show that the diagonals of the square with vertices

(0,0),(a,0),(a,a),(0,a)

have equal length.

Example 6

medium

In a coordinate proof, you compute the slopes of two segments to be

\frac{2}{3}

each. What does this prove about the segments?

Example 7

medium

For the triangle

A(0,0), B(6,0), C(2,4)

, is the median from

C

to midpoint of

AB

perpendicular to

AB

Is the median from C to midpoint M of AB perpendicular to AB?

Example 8

hard

Using coordinates

(0,0),(a,0),(a,b),(0,b)

, prove the diagonals of any rectangle bisect each other.

Example 9

hard

Place a rhombus with vertices

(0,0), (a, 0), (a+c, d), (c, d)

where

a^2 = c^2 + d^2

. Show its diagonals are perpendicular.

Example 10

hard

Prove that the segment joining midpoints of two sides of triangle

(0,0), (6,0), (2,8)

is parallel to the third side.

Example 11

hard

Prove the right triangle with vertices

(0,0), (a,0), (0,b)

has its hypotenuse length

\sqrt{a^2+b^2}

Find the hypotenuse length using the distance formula.

Example 12

hard

Find the distance from

(0,0)

to the centroid of the triangle

(0,0),(6,0),(3,6)

Find the distance from A(0,0) to the centroid G(3,2) of triangle ABC.

Example 13

challenge

Using coordinates, prove that the three medians of triangle

(0,0),(6,0),(3,6)

are concurrent at the centroid.

Example 14

challenge

Using coordinates, prove that in any triangle, the perpendicular from a vertex to the opposite side lies inside the triangle iff that vertex's angle is acute.

← Back to Coordinate Proofs Practice Mode

Related Concepts

Distance Formula Midpoint Formula Slope in Geometry Analytic Geometry

Background Knowledge

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

distance formulamidpoint formulaslope in geometry