Segments in Circles
Segments in Circles
Objectives
Students will solve problems involving segment lengths within circles. Students will:
- discover the segment product properties of circles through the use of technology.
- apply the perpendicular property of tangents and the Pythagorean theorem to find missing segment lengths.
- apply theorems and formulas involving secant, tangent, and chord segments to solve problems involving algebra.
Essential Questions
- What are the different characteristics of circles and how can they be used to solve problems? [IS.4 - All Students]
Vocabulary
[IS.1 - All Students] [IS.2 - All Students]
- Chord: A line segment whose endpoints are on a circle. [IS.3 - All Students]
- Circumscribed Polygon: A polygon such that every side of the polygon is tangent to the curve and that the curve is contained in the polygon.
- Converse of the Pythagorean Theorem: If in a triangle, a2 + b2 = c2 and a, b, and c are the sides of the triangle, then the triangle is a right triangle; if c2 > a2 + b2, then the triangle is an obtuse triangle; if c2 < a2 + b2, then the triangle is an acute triangle.
- Inscribed Polygon: A polygon such that every vertex of the polygon is incident upon the curve and the polygon is contained inside the curve.
- Intersecting Chord Theorem: If two chords intersect in a circle, then the products of the lengths of the chords segments are equal.
- Perpendicular Lines: Two lines, segments, or rays that intersect to form right angles.
- Pythagorean Theorem: The sum of the squares of the lengths of the legs of a right triangle is equal to the square of the length of the hypotenuse; in any right triangle where the length of one leg is a, the length of the second leg is b, and the length of the hypotenuse is c, as in: c2 = a2 + b2.
- Secant (of a circle): A line that intersects a circle in exactly two points.
- Secant Segment Theorem: If two secants intersect in the exterior of a circle, then the product of the measure of one secant segment and its external secant segment is equal to the product of the measures of the other secant and its external secant segment.
- Segment: A part of a line with two endpoints.
- Tangent (of a circle): A line that touches a circle in exactly one point. Tangent theorem 1 states that a line is tangent to a circle if and only if it is perpendicular to a radius drawn to the point of tangency. Tangent theorem 2 states that if two line segments from the same exterior point are tangent to the same circle, then they are congruent.
- Tangent Secant Segment Theorem: If a tangent and a secant intersect in the exterior of a circle, then the square of the measure of the tangent is equal to the product of the measures of the secant segment and its external secant segment.
Duration
8 hours–8 periods
Prerequisite Skills
Prerequisite Skills haven't been entered into the lesson plan.
Materials
- Lesson 3 PowerPoint presentation (M-G-6-3_Lesson 3 PowerPoint.pptx)
- Printout of slides 3, 4, 5, 7, 8, 10, 11, 13, 15 for students from the Lesson 3 PowerPoint presentation
- Concept Builder worksheet (M-G-6-1_Concept Builder.xls)
Related Unit and Lesson Plans
Related Materials & Resources
The possible inclusion of commercial websites below is not an implied endorsement of their products, which are not free, and are not required for this lesson plan.
- This site provides more examples and guided-practice problems covering the segment product theorems from Lesson 3, and could be used for extra practice.
http://www.mathwarehouse.com/geometry/circle/tangent-secant-side-length.php
- This site has links to a PDF file containing an investigation lab for the secant segment theorem provided in Lesson 3. This partner activity allows students to discover a relationship, rather than just be told a formula:
- Extra practice and examples to cover the tangent theorem, discussed as theorem 1 in Lesson 3.
http://www.mathwarehouse.com/geometry/circle/tangent-to-circle.php.
- Hands-on activity for discovering that two tangents drawn to the same external point from a circle are equal.
- Calculator files for the Technology Explorations
- TI-84 calculators:
http://education.ti.com/educationportal/activityexchange/Activity.do?cid=US&aId=12512
- TI-Nspire™ calculators:
http://education.ti.com/educationportal/activityexchange/Activity.do?cid=US&aId=12513
Formative Assessment
Suggested Instructional Supports
Instructional Procedures
Related Instructional Videos
Note: Video playback may not work on all devices.
Instructional videos haven't been assigned to the lesson plan.
DRAFT 10/13/2011