Precise Geometric Definitions
Aligned to G-CO.A.1 — Common Core State Standards for Mathematics.
What this lesson teaches
Geometry is built on undefined notions (point, line, distance, plane) and precise definitions built from them. An angle is formed by two rays with a common endpoint; a circle is the set of all points a fixed distance from a center; perpendicular lines meet at a 90° angle. Precise definitions make proofs unambiguous.
Worked example
Define a circle using distance. A circle with center O and radius r is the set of all points P such that the distance OP equals r. So every point on the circle is exactly r units from O, and no other point is. This is why every radius of one circle has the same length.
Practice questions
- Write a precise definition of a line segment using the words point and endpoint.
- Two lines are called perpendicular. Complete the definition: they intersect to form a ______ angle.
- Explain why 'a circle is a round shape' is not a precise mathematical definition.
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
Definiciones geométricas precisas
La geometría se construye sobre nociones no definidas (punto, línea, distancia, plano) y definiciones precisas basadas en ellas. Un ángulo se forma con dos rayos que comparten un extremo; un círculo es el conjunto de puntos a una distancia fija de un centro; las líneas perpendiculares se cortan en un ángulo de 90°. Las definiciones precisas hacen inequívocas las demostraciones.
Ejemplo: Define un círculo usando la distancia. Un círculo con centro O y radio r es el conjunto de todos los puntos P tales que la distancia OP es igual a r. Así, cada punto del círculo está exactamente a r unidades de O, y ningún otro lo está. Por eso todo radio de un círculo tiene la misma longitud.
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