Coordinate Proofs
Aligned to G-GPE.B.4 — Common Core State Standards for Mathematics.
What this lesson teaches
Placing a figure on the coordinate plane lets you prove geometric facts with algebra. Use the distance formula for side lengths, slope for parallel or perpendicular sides, and the midpoint formula for bisected segments. Choosing convenient coordinates (a vertex at the origin) simplifies the algebra.
Worked example
Prove the diagonals of a rectangle are equal. Place vertices at A(0,0), B(a,0), C(a,b), D(0,b). Diagonal AC has length sqrt((a-0)^2 + (b-0)^2) = sqrt(a^2 + b^2). Diagonal BD has length sqrt((0-a)^2 + (b-0)^2) = sqrt(a^2 + b^2). The two lengths are equal, so the diagonals of any rectangle are congruent.
Practice questions
- Use the distance formula to show the triangle with vertices (0,0), (4,0), (0,4) is isosceles.
- Prove the segment joining midpoints of two sides of the triangle (0,0), (6,0), (0,8) is half the third side using coordinates.
- Place a square with vertices (0,0), (s,0), (s,s), (0,s) and show its diagonals are equal in length.
Watch the lesson
Every lesson comes with a video taught in English and Spanish — the same video the QR code in the printed workbook opens.
▶ Watch this lessonEn español
Demostraciones con coordenadas
Colocar una figura en el plano coordenado permite demostrar hechos geométricos con álgebra. Usa la fórmula de distancia para longitudes de lados, la pendiente para lados paralelos o perpendiculares, y la fórmula del punto medio para segmentos bisecados. Elegir coordenadas convenientes (un vértice en el origen) simplifica el álgebra.
Ejemplo: Demuestra que las diagonales de un rectángulo son iguales. Coloca los vértices en A(0,0), B(a,0), C(a,b), D(0,b). La diagonal AC mide raíz((a-0)^2 + (b-0)^2) = raíz(a^2 + b^2). La diagonal BD mide raíz((0-a)^2 + (b-0)^2) = raíz(a^2 + b^2). Las dos longitudes son iguales, así que las diagonales de cualquier rectángulo son congruentes.
More Expressing Geometric Properties with Equations lessons
Teach this lesson today
An AI tutor that guides with questions instead of just giving answers — all K–12 core subjects, every child in the family.
Start free — no card required