Coordinate Quadrilateral Proofs
A free Geometry lesson from the “Coordinate Geometry and Proof” unit, with a worked example and practice problems including step-by-step solutions.
A coordinate proof converts geometric claims into computations whose meaning is stated. Equal slopes establish parallel nonvertical lines; slopes whose product is -1 establish perpendicular nonvertical lines. Squared distances compare lengths without rounding. A classification needs enough properties to force the conclusion, and the vertices must be considered in their stated order.
What you'll learn
- Use slope to prove parallel and perpendicular sides
- Use distance to prove congruent sides
- Classify quadrilaterals on the coordinate plane
Worked example
Problem. For vertices A(0,0), B(4,2), C(3,4), D(-1,2) in order, prove ABCD is a rectangle and decide whether it is a square.
- The slopes of AB and CD are 2/4 = 1/2 and (-2)/(-4) = 1/2. The slopes of BC and DA are 2/(-1) = -2 and (-2)/1 = -2.
- Both pairs of opposite sides are parallel, so ABCD is a parallelogram.
- Adjacent slopes have product (1/2)(-2) = -1, so an angle is right. A parallelogram with a right angle is a rectangle.
- AB^2 = 4^2 + 2^2 = 20, while BC^2 = (-1)^2 + 2^2 = 5. Adjacent sides are unequal, so it is not a square.
Answer: ABCD is a rectangle but not a square.
Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (multiple choice and typed response). Counts include repeated prompts and variants.
Practice problems
1. Equal slopes prove lines are...
Choices: Parallel · Perpendicular · Congruent · Supplementary
Show solution
- Equal slopes mean parallel lines if distinct.
Answer: Parallel
2. Slopes 3 and -1/3 prove lines are...
Choices: Perpendicular · Parallel · Neither · Horizontal
Show solution
- They are opposite reciprocals.
Answer: Perpendicular
3. Both pairs of opposite sides parallel proves a quadrilateral is a...
Choices: Parallelogram · Circle · Triangle · Ray
Show solution
- This is the parallelogram definition.
Answer: Parallelogram
4. Four congruent sides prove a quadrilateral is at least a...
Choices: Rhombus · Rectangle · Trapezoid · Line
Show solution
- A rhombus has all sides congruent.
Answer: Rhombus
5. A parallelogram with perpendicular adjacent sides is a...
Choices: Rectangle · Rhombus always · Trapezoid · Pentagon
Show solution
- Perpendicular adjacent sides create right angles.
Answer: Rectangle
Explain and apply
Complete these tasks on paper. Use the rubric to review your reasoning with a teacher or study partner; these tasks are not automatically scored.
Write a coordinate classification
Plot A(1,1), B(7,3), C(6,6), D(0,4) in order. Prove or disprove: “ABCD is a square.” Give its most specific quadrilateral classification and find its area. Justify every claim with a calculation.
Review rubric and solution
- Both pairs of opposite slopes are computed and connected to parallelism.
- Perpendicularity is justified using adjacent slopes.
- Adjacent squared lengths distinguish a rectangle from a square.
- Area uses perpendicular side lengths and includes square units.
- AB and CD have slope 1/3; BC and DA have slope -3. Thus both pairs of opposite sides are parallel.
- The adjacent slope product is -1, so ABCD is a rectangle.
- AB^2 = 6^2 + 2^2 = 40 and BC^2 = (-1)^2 + 3^2 = 10. These are unequal, disproving the square claim.
- The perpendicular side lengths are sqrt(40) and sqrt(10); area = sqrt(400) = 20 square units.
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