Chapter 6; Circles

Chapter 6 – Circles

Lesson 6.1 The Circle (Textbook pages 414-421, 3 sessions)

I.          Mathematical Concepts and Skills

A.        A secant is a line that intersects the circle in two points.

B.        A tangent is a line that is coplanar with the circle and intersects the circle in exactly one point.

C.        If a radius is perpendicular to a chord, then it bisects the chord.

D.        If a radius of a circle bisects a chord that is not a diameter, then it is perpendicular to the chord.

E.        The perpendicular bisector of a chord passes through the center of the circle.

F.         Congruent circles are circles that have congruent radii.

G.        Concentric circles are coplanar circles with the same center.

H.        If chords of a circle or of congruent circles are equidistant from the center(s), then the chords are congruent.

I.          If chords of a circle or of congruent circles are congruent, then they are equidistant from the center of the circle.

II.          Objectives

            A.        Apply the definitions and concepts related to circles

B.        Prove and apply the theorems that show the relationship between radius and chord.

C.        Prove and apply the theorem about the perpendicular bisector of a chord.

D.        Identify congruent and concentric circles.

E. Prove and apply the theorems about chords that are equidistant from the center of the circle.

III.         Values Integration

            Love and concern for the other people

IV.        Materials

            Compact disc

            Manila paper

            Markers

V.        Instructional Strategy

            Discussion

VI.        Procedure

Session 1

A.        Preliminary Activities

                        1. Check the assignment

2. Recall the definitions of circles, radius, chord and diameter.

B.        Lesson Proper         

1.         Bring a compact disc. Use it as a springboard to the lesson. Tell the students that a CD is 12.5 cm in diameter but the track where the information is coded is thinner than hair that it rotates at a speed of 200 rpm.

2.         Ask the students to give some more examples of objects that suggest circles.

3.         Ask the students why a ring is used to represent love.

4.         Draw the figure and ask the students to identify the radius, chord, and diameter. Then introduce the terms secant and tangent. Ask the students to compare line EF and line GH. Introduce also the terms interior and exterior of a circle. Ask the students to compare the distance of point I as well as the distance of point J from O with the length of the radius.

5.         Guide the students in proving Theorems 6-1 and 6-3.

            6.         Practice Exercises

                        a. Prove Theorem 6-2

                        b. Exercises

Solve Written Mathematics of Exercise 6.1 (numbers 1 and 2) on page 423 of the textbook.

           

            C.        Assignment

Solve Written Mathematics of Exercise 6.1 (numbers 3 and 4) on page 423 of the textbook.

Session 2

A.        Preliminary Activities

            1. Check the assignment

            2. Recall the Hypotenuse-Leg Theorem

B. Follow-up Lesson

1. Introduce the concept of congruent circles by considering the following figures

            Ask the students which of the above circles are congruent. Ask them to define congruent circles in their own words.

2. Introduce to the class concentric circles by drawing on the board a figure similar to the one below.

3. Activity

a. Draw a circle

b. Draw two chords equidistant from the center.

c. Carefully measure the lengths of two chords.

d. What do you notice?

                        4.   Guide the students in proving Theorem 6-4 deductively.

5.   Ask the students to state the converse of Theorem 6-4. Then ask them to prove Theorem 6-5

6.   Practice exercises

a.   Answer Mental Mathematics of Exercise 6.1 (numbers 1-10) on page 422 of the textbook.

b.   Solve Written Mathematics of Exercise 6.1 (numbers 5 and 6) on page 423 of the textbook.

C. Assignment

Solve the Written Mathematics of Exercise 6.1 numbers (7 and 8) on page 424 of the textbook.

Session 3

A. Preliminary Activity

      Check up the assignment.

B. Follow-up Lesson

      1. Illustrative Example

            Explain the proof of this problem on the board.

            Given: Radius OA and DC

                        D is the midpoint of chord AC

            Prove: OB is perpendicular bisector of AC

2. Practice Exercise

Solve Written Mathematics of Exercise 6.1 (numbers 9 and 10) on page 424 of the textbook.

C. Checking the Understanding.

            Solve Written Mathematics of Exercise 6.1 (numbers 11-15) on pages 424-425 of the textbook.

D. Assignment

Solve Written Mathematics of Exercise 6.1 (numbers 16-20) on pages 424-425 of the textbook.

If there is enough time, solve Challenge problems of Exercise 6.1 on page 426 of the textbook.

Lesson 6.2 - Arcs and Angles (Textbook pages 427-446, 3 sessions)

I.          Mathematical Concepts and Skills

A. A central angle is an angle whose vertex is the center of the circle.

B. Minor arc AB is the union of two points A and B and all the points of the circle   in the interior <AOB

C. Major arc ACB is the union of points Q and B and all points of the circle in the Exterior of Central angle <AOB.

D. A semicircle is the union of the endpoints of a diameter and all points of the circle that lie on one side of the diameter.

E. The degree measure of a minor arc is equal to the degree measure of the central angle.

F. The degree measure of a major arc is equal to 360 degrees minus the degree measure of its related minor arc.

G. The degree measure of a semicircle is 180 degree.

H. The measure of an arc formed by two adjacent, nonoverlapping arcs is the sum of the measures of the measures of the two arcs. 

I. In the same circle of in congruent circles, arcs which have the same measures are congruent arcs.

J. If two minor arcs of a circle or of congruent circles are congruent, then the corresponding chords are congruent.

K. If two of a circle or congruent circles are congruent, then their corresponding minor arcs are congruent.

L. If two central angles of a circle or of congruent circles are congruent, then the corresponding minor arcs are congruent.

M. If two minor arcs of a circle or of congruent circles are congruent, then the corresponding central angles are congruent.

N. If two central angles of a circle or of congruent circles are congruent, then their chords are congruent.

O. If two chords of a circle or of congruent circles are congruent, then the central angles are congruent.

II.          Objectives

            A. To illustrate and define the following:

                        1. central angle                                  2. Minor arc

3. Major arc                                        4. Semicircle

5. Degree measure of minor arc     6. Degree measure of major arc

7. Degree measure of semicircle   8. Congruent arc

B. To illustrate and apply the Arc addiction postulate.

C. To prove and apply the theorems about arcs and central angles.

III.         Values Integration

            Friendship

IV.        Materials

            Manila paper

            Marker

                                        

V.        Instructional Strategy

            Whole Class Discussion

VI.        Procedure

            Session 1

A. Preliminary Activity

Check the assignment.

B. Lesson Proper

      1. Introduce the following terms. Use the figure on page 427 of the textbook.

            a. Central Angle                                b. Minor Arc

c. Major Arc                                       d. Semicircle

e. Degree measure of minor arc     f. Degree measure of major arc

g. Degree measure of semicircle   h. congruent arc

2. Illustrate how to use the Arc Addition Postulate

3. Associate Arc Addition Postulate with making friends with other students.

4. Guide the students in proving Theorem 6-6

5. Practice Exercises

      a. Prove Theorem 6-7

b. Prove Theorem 6-8

c. Prove Theorem 6-9

C. Assignment

      Prove Theorems 6-10 and 6-11. Refer to pages 431-432 of the textbook.

Session 2

A.        Preliminary Activity

            Check the assignment

B.        Follow-up Lesson

            1. Draw the following figures on the board.

Ask the students to compare <AOB of Circle O and <ABC of circle O. After making it clear with the students that <ABC is an inscribed angle, ask the students to define an inscribed angle.

2. Copy the figure on the board:

Ask the students to measure <ABC and <APC. Ask the students to write a formula for m<ABC in terms of <APC. Then ask them to write a formula for m<ABC in terms of arc AC.

3. Guide the students in proving Theorem 6-12.

4. Illustrative Example

Use the example on page 442 of the textbook.

5. Practice Exercises

 a. Answer Mental Mathematics of Exercise 6.2 (numbers 1-10) on page 442 of the textbook.

b.   Solve Written Mathematics of Exercise 6.2 (numbers 1-4) on page 443 of the textbook.

C. Assignment

Solve Written Mathematics of Exercise 6.2 (numbers 5-10) on page 443 of the textbook.

Session 3

A.        Preliminary Activities

1. Check the assignment

2. Recall Theorem 6-12 and the definition of a right angle.

B.        Follow-up Lesson

1.   a.   Guide the students in proving Corollary 6-12.1.

            An angle inscribed in a semicircle is a right angle.

      b. Explain the proof of Corollary 6-12.4.

      If two arcs of a circle are inscribed between parallel secants, then the arcs are congruent.

2.   Practice Exercises

      a. Prove the following:

            Corollary 6-12.2

            If two inscribed angles intercept the same arc or congruent arcs, then the angles are congruent.

      Corollary 6-12.3

            Opposite angles of an inscribed quadrilateral are supplementary.

b. Solve Written Mathematics of Exercise 6.2 (numbers 11-14) on pages 443-444 of the textbook.

C.        Checking for understanding

            Solve Written Mathematics of Exercise 6.2 (numbers 15-20) on pages 443-444 of the textbook.

D.        Assignment

            Solve Written Mathematics of Exercise 6.2 (numbers 21-28) on pages 443-445 of the textbook.

            If there is enough time, solve for the challenge problem of Exercise 6.2 on page 446 of the textbook.

Lesson 6.3 - Tangent lines and Tangent Circles (Textbook pages 447-459, 4 sessions)

I. Mathematical Concepts and Skills

A. If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.

B. If a line is perpendicular to a radius at its outer endpoint, then it is tangent to the circle.

C. If two segments from the same external point are tangent to a circle then the two segments are congruent.

D. Two circles are tangent to each other, if and only if they are coplanar and are tangent to the same line at the same point.

E. Two circles are externally tangent, if and only if each lies in the exterior of the other except for the point of tangency.

F.   Two circles are internally tangent, if and only if one of them lies in the interior of the other except for the point of tangency.

G. If two circles are tangent, then their centers and their point of tangency are collinear.

H.  A common tangent is a line that is a tangent to each of two coplanar circles.

I.    A common internal tangent is a common tangent that intersects the segment joining the centers of two circles.

J.   A common external tangent is a common tangent that does not intersect the segment joining the centers of two circles.

II.    Objectives

A. To prove and apply the theorems about tangent lines to circle

B.  To define the following terms:

      a.   tangent circles                                   b.   externally tangent circles

      c.   internally tangent circles                   d.   common tangent

      e.   common internal tangent                  f.    common external tangent

C. To prove if two circles are tangent, then their centers and the point of tangency are collinear.

III.   Values Integration

      Concern for others

IV.  Materials

      Activity sheets

V. Instructional Strategies

      A.  Whole class discussion

      B. Small group discussion

VI. Procedure

      Session 1

      A.  Preliminary Activities

            1. Check the assignment

            2. Recall indirect proof.

      B. Lesson Proper

            1.   Activity

                  a.   Draw a circle. Name its center O.

                  b.   Draw a line tangent to the circle at point A.

                  c.   Draw OA. What part of circle is OA?

                  d.   Using your protractor, determine whether the tangent is perpendicular to                                 OA. What can you conclude?

            2.   Guide the students in proving Theorem 6-13 and using indirect proof.

            3.   Practice Exercises

                  a.   Divide the class into smaller groups. Assign the best in the class to be the leaders in each group. Tell them to help those who find proving geometric statements difficult. Tell them also that showing concern for other people is a virtue that they should possess as future leaders. Ask the class to prove Theorem 6-14.

                  b.   Answer Mental Mathematics of Exercise 6.3 (numbers 1-10) on pages 454-455 of the textbook.

                  c.   Solve Written Mathematics of Exercise 6.3 (numbers 1-4) on page 455 of the textbook.

      C. Assignment

                                    Solve Written Mathematics of Exercise 6.3 (numbers 5 and 6) on page 456 of the textbook.

Session 2

A. Preliminary Activities

      1.   Check the assignment

      2.   Recall the Hypotenuse-Leg Theorem.

B.  Follow-up Lesson

      1.   Guide the students in proving Theorem 6-15

      2.   Illustrative Example

            AB and BC are tangent to circle O at A and C, respectively. If BA=12 and OA=9, find BC, and OB.

3.   Practice Exercises

      Solve Written Mathematics of Exercise 6.3 (numbers 7 and 8) on page 456 of the textbook.

      C. Assignment

Solve Written Mathematics of Exercise 6.3 (numbers 7 and 8) on page 456 of the textbook.

Session 3

A. Preliminary Activities

      1.   Check the assignment.

      2.   Recall the definition of collinear points.

B.  Lesson Proper

      1.   Arrange the students in pairs. Then ask them to do the following activities:

            Activity 1

            a.   Draw a circle. Name the center P.

            b.   Draw a line tangent to circle P at any point R on the circle.

            c.   Draw another circle tangent to the same line at same point of tangency. In how many ways can this be done?

            d.   Draw a line through the center of the circles and the common point of tangency. What do you observe?

      2.   After doing Activity 1, ask the students to define tangent circles, externally tangent circles and internally tangent circles.

      3.   Discuss the proof of Theorem 6-16.

            Activity 2

            a.   Draw a circle. Name the center of your circle O.

            b.   Draw another circle that does not intersect the first one. Name the center P.

            c.   Join the centers of the two circles by a segment.

            d.   Draw a line tangent to both circles that:

·        Intersect the segment joining the centers of the two circles

·        Does not intersect the segment joining the centers of the two circles.

            4.   After doing Activity 2, ask the students to define common tangent, common internal tangent, and common external tangent.

            5.   Illustrative Example

                  Use the example on page 453 of the textbook.

            6.   Practice Exercise

                        Solve Written mathematics of Exercise 6.3 (numbers 11 and 12) on page 457 of the textbook.

C. Assignment

            Solve Written Mathematics of Exercise 6.3 (numbers 13 and 14) on page 457 of the textbook.

Session 4

A.  Preliminary Activities

      Check the assignment

B. Follow-up Lesson

      1.   Illustrative Example

            Explain the proof of this problem.

            Given: Secant AE and DE intersecting at B and C.

            Prove: Triangle ACE is similar to Triangle DBE

     

2.   Practice Exercises

                  Solve Written Mathematics of Exercise 6.3 (numbers 15 and 16) on page 458 of the textbook.

C. Checking for Understanding

                  Solve Written Mathematics of Exercise 6.3 (numbers 17 and 18) on page 458 of the textbook.

D. Assignment

                  Solve Written Mathematics of Exercise 6.3 (numbers 19 and 20) on page 458-459 of the textbook.

If there is enough time, solve Challenge problems of exercise 6.3 on page 459 of the textbook.

Lesson 6.4 Angles formed by Tangents and Secants

I.    Mathematical Concepts and Skills

      A. If two secants intersect in the interior of a circle, then the measure of the angle formed is equal to one-half of the sum of the measures of the intercepted arcs.

      B. If two secants intersect in the exterior of a circle, then the measure of the angle formed is one-half the positive difference of the measures of the intercepted arcs.

      C. The measure of an angle formed by a secant and a tangent intersecting in the exterior of a circle is one-half the positive difference of the measures of the intercepted arcs.

      D. The measure of an angle formed by a secant and a tangent intersecting in the exterior of a circle is one-half the positive difference of the measures of the intercepted arcs.

      E.  The measure of an angle formed by two tangents to the same circle is one-half the positive difference of the measures of the intercepted arcs.