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Proving Theorems with
Coordinate Geometry

Use coordinates, formulas, and algebra to prove geometric relationships: distances, midpoints, slopes, and the properties of shapes on the plane.

Geometry

What Is a Coordinate Proof?

A coordinate proof places geometric figures on the coordinate plane and uses algebraic formulas to verify properties. Instead of relying on angle postulates alone, you compute distances, slopes, and midpoints to prove a claim.

Your Formula Toolkit

Distance
\( d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2} \)
Midpoint
\( M = \left(\frac{x_1+x_2}{2},\,\frac{y_1+y_2}{2}\right) \)
Slope
\( m = \frac{y_2-y_1}{x_2-x_1} \)

Why Use Coordinates to Prove?

1
Precision: Exact numerical results eliminate ambiguity: either the numbers match or they don't.
2
Versatility: Works for any shape: triangles, quadrilaterals, circles, and more.
3
Algebra bridge: Connects geometry to algebra, building skills for analytic geometry and calculus.

Slope Rules: Parallel vs. Perpendicular

\( m_1 \)
\( m_2 \)
Parallel: \( m_1 = m_2 \)
\( m_1 \)
\( m_2 \)
Perpendicular: \( m_1 \cdot m_2 = -1 \)

Proving Points on a Circle

x y (0, 2)
\( (1, \sqrt{3}) \)
 (0, 0)
\( r=2 \)

Prove vs. Disprove: Know the Difference

To Prove
Show that every required property holds. All distances match, all slopes satisfy the condition, all midpoints align: every check must pass.
To Disprove
Find one property that fails. A single mismatched distance, a slope that isn't perpendicular, or a point off the curve: one counterexample is enough.

How to Write a Coordinate Proof

1
Plot the points. Place all given vertices on the coordinate plane and label them.
2
Identify what to prove. Is it a rectangle? Parallelogram? Point on a circle? List the required properties.
3
Choose formulas. Select distance, midpoint, or slope depending on the property.
4
Calculate and compare. Plug in coordinates, simplify, and check all conditions.
5
State your conclusion. Clearly say whether the theorem is proved or disproved, citing the evidence.

Example 1: Point on a Circle

Prove or disprove: The point \( (1, \sqrt{3}) \) lies on the circle centered at the origin that contains \( (0, 2) \).

1
Find the radius. The circle passes through \( (0,2) \), so \( r = \sqrt{0^2 + 2^2} = 2 \).
2
Write the equation. \( x^2 + y^2 = 4 \).
3
Test the point. Substitute \( (1, \sqrt{3}) \): \( 1^2 + (\sqrt{3})^2 = 1 + 3 = 4 \). ✓

Example 2: Is ABCD a Rectangle?

Given: \( A(0,0) \), \( B(4,0) \), \( C(4,3) \), \( D(0,3) \). Prove that \( ABCD \) is a rectangle.

\( AB = 4 \)
\( BC = 3 \)
\( CD = 4 \)
\( DA = 3 \)
 A B C D

Example 2: Completing the Proof

1
Opposite sides equal: \( AB = CD = 4 \), \( BC = DA = 3 \). So \( ABCD \) is a parallelogram.
2
Adjacent sides perpendicular: Slope of \( AB = 0 \) (horizontal), slope of \( BC \) is undefined (vertical). A horizontal and vertical line meet at \( 90^\circ \).
3
Diagonals equal: \( AC = \sqrt{4^2+3^2} = 5 \), \( BD = \sqrt{(0-4)^2+(3-0)^2} = 5 \). ✓

Example 3: Midpoint Method

Prove: The diagonals of parallelogram \( P(1,1) \), \( Q(5,1) \), \( R(7,4) \), \( S(3,4) \) bisect each other.

(4, 2.5) P(1,1) Q(5,1) R(7,4) S(3,4)

What to Check for Each Quadrilateral

Shape Key Properties Formulas Used
Parallelogram Opposite sides equal OR parallel Distance or Slope
Rectangle Parallelogram + right angles Slope (perpendicular)
Rhombus All four sides equal Distance (4 sides)
Square Rhombus + right angles Distance + Slope
Trapezoid Exactly one pair of parallel sides Slope (2 pairs)

Example 4: Disproving a Rectangle

Disprove: \( A(0,0) \), \( B(5,0) \), \( C(6,3) \), \( D(1,3) \) form a rectangle.

1
Check slopes of adjacent sides:
Slope of \( AB = \frac{0-0}{5-0} = 0 \).
Slope of \( BC = \frac{3-0}{6-5} = 3 \).
Slope of \( AD = \frac{3-0}{1-0} = 3 \).
2
Test perpendicularity: For \( AB \perp BC \), we need \( 0 \times 3 = 0 \neq -1 \). ✕ Fails!
3
Conclusion: Adjacent sides are not perpendicular. This is a parallelogram (since \( AB \parallel CD \) and \( BC \parallel AD \)), but not a rectangle.

Common Mistakes to Avoid

✕ Forgetting to Check All Conditions
Showing equal sides alone doesn't prove a rectangle: you must also verify right angles.
✕ Slope of a Vertical Line
Vertical lines have an undefined slope. Don't write \( m = 0 \): that's horizontal.
✕ Arithmetic Errors in Distance
Forgetting to square the differences, or misplacing a negative sign under the radical.

Question 1 of 5

Does the point \( (3, 4) \) lie on the circle centered at the origin with radius \(5\)?

Question 2 of 5

What is the distance between \( A(1, 2) \) and \( B(4, 6) \)?

Question 3 of 5

Lines \( \ell_1 \) and \( \ell_2 \) have slopes \( m_1 = 2 \) and \( m_2 = -\frac{1}{2} \). Which statement is true?

Question 4 of 5

To prove that quadrilateral \( ABCD \) is a rhombus, which property is sufficient?

Question 5 of 5

The midpoint of segment \( \overline{PQ} \) where \( P(-2, 3) \) and \( Q(6, -1) \) is: