Circle Conjectures
Project AMP Dr. Antonio R. Quesada – Director, Project AMP
Circle Conjectures
Lab Summary: This lab will deal with certain properties of circles. In particular, eight conjectures will be covered in this lab. As each problem of the lab is completed, the student should discover certain properties that hold for circles. These properties relate to tangents lines, inscribed angles, and chords of circles. Upon completion if this lab, the student should be familiar with all these concepts. The lesson plan I have implemented involves conjectures of circles. These conjectures are certain properties on circles that the students will discover through working with the lab. Included in this lesson plan is:
· A statement of the objectives this lesson will cover, that is, the conjectures to be discovered by the students. Each conjecture is defined and a diagram of each is provided. This is the knowledge the students are to have learned after lesson is complete. The teacher can use this to review the conjectures once the students have completed the labs to ensure that the students have a firm grasp of the material. This can also be used as a guide when grading the labs.
· A list of definitions needed in order to complete the lab. These definitions are meant to be a review of material the student should have seen already. If needed, the teacher may want to briefly go over each of these terms to ensure that the students have a firm grasp on this material before starting the lab. The teacher could also provide each student with a copy of the definitions to be used as a reference when working on the lab.
· A lab in which the students will explore through Cabri the conjectures to be covered in this lesson. The lab will help the students to discover for themselves each conjecture that is to be covered. By completing each lab problem, the student will learn the conjecture covered by that problem. The lab is a lesson plan in and of itself, in that, in order to complete the lab, the student must discover each conjecture. Therefore, it is up to the teacher to decide how much review is needed before starting the lab, and how much explanation is needed after the lab is completed to ensure the concepts have been grasped. Two extension questions are also provided in order to test and challenge what the students have learned from the lab. Evaluating their lab work will provide an evaluation of the students’ understanding of the material. Successful completion of the lab should result in the students having a firm grasp of the material.
Key Words: Circles, tangents, radius, inscribed, perpendicular, chords, tangent segments
Project AMP Dr. Antonio R. Quesada – Director, Project AMP
Background knowledge: This lesson will concentrate on certain properties found in circles. The lesson is implemented through a lab in which the student will discover each property through experimentation using Cabri. Specifically, the conjectures covered will be:
· Tangent Conjecture I: Any tangent line to a circle is perpendicular to the radius drawn to the point of tangency.
· Tangent Conjecture II: Tangent segments to a circle from a point outside the circle are equal in length.
· Inscribed Angles Conjecture I: In a circle, the measure of an inscribed angle is half the measure of the central angle with the same intercepted arc.
Project AMP Dr. Antonio R. Quesada – Director, Project AMP
· Inscribed Angles Conjecture II: In a circle, two inscribed angles with the same intercepted arc are congruent.
· Inscribed Angles Conjecture III: Any angle inscribed in a semi-circle is a right angle.
· Congruent Chords Conjecture: If two chords are congruent, then the following properties hold: 1. They determine central angles that are equal in measurement. 2. Their intercepted arcs are congruent. Also, the chords are equal distance from the center.
· Perpendicular Bisector of a Chord Conjecture: The perpendicular bisector of a chord in a circle passes through the center of the circle.
Project AMP Dr. Antonio R. Quesada – Director, Project AMP
Learning Objectives: 1. Students will discover the eight conjectures covered in background knowledge.
Materials: Geometry software Worksheets
Procedures: Each of these conjectures is covered as a separate problem in the lab that follows. The student should be able to discover these conjectures themselves by working out the problems using Cabri. Two extension questions are given which challenge the student to put to use certain conjectures discovered earlier in order to come up with a solution. Extension I requires students to use the perpendicular bisector of a chord property in order to find the center of any given circle or arc. To do this, the student should construct two chords in the given circle or arc and bisect each with a perpendicular line. Where the two perpendicular lines intersect is the center of the circle. Extension II requires students to use the method found in Extension I to, given any arc, find the circumference of the circle the arc is a part of. To do this student should find the center of the circle using the method discovered in Extension I. Once the center is found, construct a radius and measure it. Use this measurement to find the circumference of the circle.
Assessment: Completed worksheets
Project AMP Dr. Antonio R. Quesada – Director, Project AMP
Circle Conjectures Activity One: Tangent and Radii
Team members’ names: ______
File name: ______
Goal: Investigate the angle formed by the tangent and the radius of a circle. a. Construct a circle A with radius AB . [use circle tool/segment tool] b. Construct a point, C on the circle and draw a secant that runs through B and C (see Fig. 1). [use point/line tool] c. Measure the angle Ð ABC that is formed. [use angle tool] suur d. Now grab and move point C around the circle until the line BC is tangent to the circle. What is the measurement of angle Ð ABC now?
What conjecture can you make about a tangent and a radius to the point of tangency?
e. Use this conjecture to construct a tangent line to a circle at a given point P.
Fig. 1
Project AMP Dr. Antonio R. Quesada – Director, Project AMP
Circle Conjectures Activity Two: Tangent Segments
Team members’ names: ______
File name: ______
Goal: Investigate the relationship between tangent segments.
a. Construct a circle A with radius AB . [use circle and segment tools] b. Construct a tangent line, t through B. [use perpendicular line tool] c. Construct another radius AD and a tangent line, s through C(see Fig2). [use point, segment, and line tools] d. Construct and label the point where lines t and s intersect as point D. [use intersection points and label tool] e. Measure the tangent segments DB and DC , what relationship exists? [use distance tool]
f. Grab and move point B around the circle. What happens to the segment measures of DBand DC ?
What conjecture can you make about tangent segments from the same exterior point?
Fig. 2
Project AMP Dr. Antonio R. Quesada – Director, Project AMP
Circle Conjectures Activity Three: Central and Inscribed angles
Team members’ names: ______
File name: ______
Goal: Investigate the relationship between inscribed and central angles.
a. Construct a circle A. b. Construct an inscribed angle Ð BCD. c. Construct a central angle Ð BAD (see Fig. 3). d. Measure both angles and compare. e. Construct diameter DE f. Drag point B around the arc DBE. What happens to the angles measures?
g. What conjecture can you say exists between central and inscribed angles that share the same intercepted arc? h. Does your conjecture remain if you drag point B pass point E? Explain why.
Fig. 3
Project AMP Dr. Antonio R. Quesada – Director, Project AMP
Circle Conjectures Activity Four: Congruent Inscribed Angles
Team members’ names: ______
File name: ______
Goal: Investigate the relationship between congruent inscribed angles.
a. Construct a circle A. b. Construct two inscribed angles, Ð BCD and Ð BED, having them share the intercepted arc BD (See Fig. 4). c. Measure the angles and compare. d. Drag point B around the arc CDE, what happens to the angle measurements? e. What conjecture can you say exists between inscribed angles that share an intercepted arc? f. What happens when you drag point B pass C to the arc CE? Explain
Fig. 4
Project AMP Dr. Antonio R. Quesada – Director, Project AMP
Circle Conjectures ActivityFive: Angles inscribed in a Semi-circle
Team members’ names: ______
File name: ______
Goal: Investigate angles inscribed in a semi-circle.
a. Draw a circle A with chord BC running through the center of the circle. b. Construct an inscribed angle Ð BDC that intercepts the arc BC (see fig. 5). c. Measure the angle Ð BDC. d. Drag point D around the circle, what happens to the angle measure? e. What conjecture can you say exists for an inscribed angle which intercepts a half circle?
Fig. 5
Project AMP Dr. Antonio R. Quesada – Director, Project AMP
Circle Conjectures Activity Six: Congruent Chords
Team members’ names: ______
File name: ______
Goal: Investigate the relationship between congruent chords.
a. Construct a circle A with a chord BC. b. Construct a new chord DE, which is a reflection of chord BC with respect to the center of the circle, point A (use the symmetry tool). Chords DE and BC should be congruent. Measure their lengths to make sure. c. Construct central angles Ð BAC and Ð DAE. Measure each of these angles and compare their values, what relationship exists?
d. Through experimenting, what other conjecture(s) can you find that exist between congruent chords in a circle? (Hint: examine arcs and distance from center).
Fig. 6
Project AMP Dr. Antonio R. Quesada – Director, Project AMP
Circle Conjectures Activity Seven: Radii that Bisect Chords
Team members’ names: ______
File name: ______
Goal: Investigate radii that bisect chords. a. Construct circle A with chord BC. b. Find midpoint of BC and label it M. d. Construct a radius AD through M and measure angle Ð AMC (see Fig. 7). d. Drag point B around circle, what happens to the angle measure? e. What conjecture can you make about a radius that bisects a chord?
Fig. 7
Project AMP Dr. Antonio R. Quesada – Director, Project AMP
Circle Conjectures Activity Eight: Perpendicular chords
Team members’ names: ______
File name: ______
Goal: Investigate perpendicular chords.
a. Construct a circle A with chord BC. b. Draw a line from A, perpendicular to chord BC. c. Find the intersection of line t and chord BC and label it as point P (Fig. 8). d. Measure segments PA and PB. What relationship exists?
What conjecture can you make between a chord and a line perpendicular to that chord through the center of the circle?
Fig. 8
Extension 1 Use the conjectures from above to devise a method for finding the center of any given circle or arc.
Extension 2 Given any arc, can you determine the circumference of the circle it is a part of? (Hint: use method from Extension 1 to find center of the circle).