Lesson Plan: Positions of Points, Straight Lines, and Circles with respect to Circles Mathematics

This lesson plan includes the objectives, prerequisites, and exclusions of the lesson teaching students how to find the positions of points, straight lines, and circles with respect to other circles.

Objectives

Students will be able to

  • recognize how a point relates to a circle (i.e., whether a point lies inside, outside, or on the circle) and how line segments drawn between the circle’s center and this point relate to the radius of the circle,
  • recognize how a line relates to a circle (i.e., whether it is a secant, a tangent, or outside the circle),
  • use the relationships between points and circles and between lines and circles to form and solve linear equations and inequalities,
  • understand that a tangent is perpendicular to a radius at the point of tangency and understand the converse to this: if a line is perpendicular to the diameter of a circle at one of its endpoints, then it must be a tangent to the circle,
  • recognize how two circles relate to each other,
  • find unknown lengths and angles relating to geometric constructions of circles and straight lines where
    • a circle and at least one radius are given,
    • a circle and at least one diameter are given,
    • a circle and at least one tangent are given,
    • a circle and at least one chord are given,
    • at least two circles are given and share a common center,
    • at least two circles are given and intersect,
    • at least two circles are given and touch (either on the inside or on the outside).

Prerequisites

Students should already be familiar with

  • the Pythagorean theorem,
  • solving linear equations and inequalities,
  • circle definitions,
  • intersecting chords and radii, and triangles produced by a chord and radii,
  • the perpendicular bisector of a chord.

Exclusions

Students will not cover

  • inscribed angle theorems,
  • properties of tangents to a circle from an external point,
  • angles of tangency,
  • parallel chords and tangents in a circle,
  • relationships between chords and the center of a circle,
  • central angles and arcs,
  • cyclic quadrilaterals.

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