10.2 Arcs and Chords

1. 10.2 Arcs and Chords Standard 21.0 Students prove and solve problems regarding relationships among chords, secants, tangents, inscribed and circumscribed polygons of circles

3. Measuring Arcs The measure of a minor arc, is defined to be the measure of its central angle. mGF=m<GHF=60o The measure of the major arc, is defined as the difference of 360o & associated minor arc. mGF=m<GEF=360-60=300o

4. Goal 1: Using Arcs of circles. Central angle-In a plane, an angle whose vertex is the center of a circle. Minor Arc, if the measure of the central angle is less than 180o. Major arc- if the measure of the central angle is less than 180o If the endpoints of an arc are the endpoints of the diameter, then the arc is a semicircle.

5. Naming arcs The minor arc associated with <APB is AB. The major arcs and Semicircles are named by their Endpoints and by a pt. on the Arc. The major arc is ACB.

6. 2-3 minute activity with protractor and compass Draw a circle H. Draw a diameter and lable it as EF. Find another point on the circle, label it G. Then draw a radius to this point. Find the mGF and the mGEF, mEG, and mEF . EGF is a semicircle, label it. Is there a relationship between mEG, mFG, and mEGF?

7. Arc Addition postulate The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs. mEGF=mEG+mGF

8. Example 1

9. Example 2

10. 2-3 minute activity with a partner Draw a circle, then draw two chords of the same length that start at the same point on the circle and end at other points on the circle. Label the starting point B. Label the other points of the chord that meet the circle point A and C respectively. Use tick marks to denote that the chords are congruent. Now draw a radius from the center to point A and C. Measure the arcs AB and BC. What is their relationship? Conjecture? In the same circle, or in congruent circles two minor arcs are congruent if�?

11. Chord Theorems 10.4 In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent. AB BC if and only if AB BC

12. 2-3 minutes and I will call on something? When I call on someone your partner will answer. Use a protractor and a compass. Draw a circle with a compass, draw a diameter. How do you know that it is a diameter? Draw a perpendicular chord that intersects the diameter. What did you make? Chord? What is the relationship between the 2 pieces of the chord and the diameter(use a ruler)? What is the relationship between the two pieces of the arcs that is created from the chord and split by the diameter.(hint use a protractor) Make a conjecture?If a diameter of a circle is perpendicular to the chord then�?

13. Chord Theorem 10.5 and 10.6

14. 2-3 minute activity Draw a circle E. Draw two nonintersecting chords of the same length. Label the segment of one chord AB and the other CD. Draw a line from the center to a point on the chord that is perpendicular to chord AB. Draw a line from the center to a point on the chord that is perpendicular to chord CD. Measure the lengths of each of lines that start from point E to the chord. Conjecture- If the same circle or congruent circles, two chords are congruent if and only if�

15. Theorem 10.7

16. Day 2

17. Proof of theorem 10.4 In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent. Page 610 question number 59.

18. Proof of theorem 10.6 Page 610 question number 61 Theorem 10.6 states If one chord is a perpendicular bisector of another chord, then the first chord is a diameter.