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About Circles

Pacing: This lesson could take two to four class periods

Goal: This lesson discusses numerous characteristics of circles. Inscribed polygons, equations of a circle, diameters, secants, tangents, and chords are all presented in this lesson.

Round Robin! Once students have read through this lesson, try this fun activity. Round robin tournaments are scheduled so each team players another exactly once. Circles and chords are used to schedule such a tournament.

Using a compass, have the students construct a circle that takes up most of an 8.5” \times 11” sheet of copy paper. Decide upon a collegiate conference, such as the Big 10 and instruct students to evenly space these dots around the circle. To do this, divide 360^\circ by 11. This will give students the number of degrees before each new dot is placed. Choosing 11 colored pencils, begin at one dot and draw a chord to a second dot (see diagram below). Draw parallel chords until all but one team has an opponent; the leftover team has a bye. This color represents week 1. Using a second color, start at another dot and connect it to a different team. Continue this process until all 11 weeks have been “scheduled.”

If there is an even number of teams to be scheduled, place one dot in the center of the circle and set each remaining dot on the boundary. Start by drawing a radius from the center to any point, then draw chords to the remaining teams. There will be no byes with even numbered teams.

Visualization! Fill a clear bowl or container with water. Show how concentric circles are formed by dropping a rock into water, forming ripples. Concentric circles can also be formed when raindrops hit a body of water, such as a lake, puddle, etc.

Tangent Lines

Pacing: This lesson should take one to two class periods

Goal: The purpose of this lesson is to connect the radius of a circle to its tangent. The real life applications of tangents are found in the additional examples below.

Connection! Tangents are used in Calculus; the slope of the tangent line represents the derivative. Students should make this connection visually once Calculus begins.

Additional Examples:

  1. Find the perimeter of ABCD. 50\;\mathrm{cm}

Common Tangent and Tangent Circles

Pacing: This lesson should take one class period

Goal: This lesson focuses on the properties shared with circles sharing common tangents.

Additional Example:

  1. A dirt bike chain fits snugly around two gears, forming a diagram like the one below. Find the distance between the centers of the gears. Assume the distance from B to the top of the second circle is 30.25”.\ \  30.254”

Arc Measures

Pacing: This lesson should take one class period

Goal: This lesson introduces arc measurements. Central angle knowledge is important when creating and interpreting pie charts and when determining arc length and the area of a sector.

Students may need extra practice when it comes to finding the value of the arc. Set up several questions and use personal whiteboards to check students’ understanding.

The Arc Addition Property should be familiar to students; it is quite similar to the Angle Addition Property.

Additional Example:

Identify the following in \bigodot A

A. Minor arcs

B. Major arcs

C. Semicircles

D. Arcs with equal measurement

E. Identify the four major arcs that contain point F.

Inscribed Angles

Pacing: This lesson should take one to one and one-half class periods

Goal: This lesson will demonstrate how to find measures of inscribed angles and how to find the measure of an angle formed by a tangent and a chord.

In-class activity! Have students copy the drawings below.

  1. In \bigodot A, use a protractor to find the measures of \angle CDE, \angle CFE, and \angle CBE. Determine the measure of arc CE.
    1. Write a hypothesis regarding measure arc CE and \angle CDE.
    2. Write a hypothesis regarding the measures of \angle CDE, \angle CFE, and \angle CBE.
  2. Use a protractor to measure \angle IKJ, \angle ILJ, and IHJ.
    1. Write a hypothesis about an angle whose vertex is on the boundary of a circle and whose sides intersect the endpoints of the diameter.

Angles of Chords, Secants, and Tangents

Pacing: This lesson should take one class period

Goal: This goal of this lesson is to explain the formulas for determining angle measures formed by secants and tangents.

Vocabulary: According to www.encylcopedia.com, “A secant is a line that intersects a curved surface.” A chord is a line segment connecting two points on the boundary of a circle. Thus, a chord is a segment of the secant.

Set up a chart similar to the one below to help students organize their formulas for this lesson and the next.

Example Angle Formula Segment Length
Tangent Chord \frac{1}{2} (intercepted arc)
Angles formed by two intersecting chords Sum of intercepted arcs
Angle formed by two intersecting tangents Difference of intercepted arcs
Angle formed by two intersecting secants Difference of intercepted arcs
Angle formed by a tangent and secant intersection Difference of intercepted arcs

Segments of Chords, Secants, and Tangents

Pacing: This lesson should take one class period

Goal: This goal of this lesson is to explain the formulas for determining segment lengths formed by intersecting secants and tangents.

Organize! Finish the chart began in the previous lesson.

Example Angle Formula Segment Length
Tangent Chord \frac{1}{2} (intercepted arc) N/A
Angles formed by two intersecting chords Sum of intercepted arcs

a*b = c*d

whole secant_1 * external part_1 = whole secant_2 * external part_2

Angle formed by two intersecting tangents Difference of intercepted arcs N/A
Angle formed by two intersecting secants Difference of intercepted arcs

a*b = c*d

whole secant_1 * external part_1 = whole secant_2 * external part_2

Angle formed by a tangent and secant intersection Difference of intercepted arcs

b*c = a^2

whole secant * external part = tangent^2

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CK.MAT.ENG.TE.1.Geometry.1.9

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