# Geometry Arcs and Chords. Geometry Mr. Peebles Spring 2013

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1 10.2 Arcs and Chords Geometry Mr. Peebles Spring 2013

2 Bell Ringer: Solve For r. B 16 ft. A r r 8 ft. C

3 Bell Ringer B 16 ft. Answer A r r 8 ft. C c 2 = a 2 + b 2 Pythagorean Thm. (r + 8) 2 = r Substitute values r r + 64 = r r + 64 = r = 192 r = 12 Square of binomial Subtract r 2 from each side. Subtract 64 from each side. Divide. The radius of the silo is 12 feet.

4 Daily Learning Target (DLT) Monday May 13, 2013 I can use properties of arcs and chords of circles, as applied.

5 Assignment: Pgs (1-27 Odds, 41-43) 1. 9π m m π ft cm ,394 ft (54π ) cm π yds (4 π) ft π/6 m ( π) in π ft A π/4 in G π in m cm 2

6 Using Arcs of Circles In a plane, an angle whose vertex is the center of a circle is a central angle of the circle. If the measure of a central angle, APB is less than 180, then A and B and the points of P major arc C central angle P A B minor arc

7 Using Arcs of Circles in the interior of APB form a minor arc of the circle. The points A and B and the points of P in the exterior of APB form a major arc of the circle. If the endpoints of an arc are the endpoints of a diameter, then the arc is a semicircle. major arc C central angle P A B minor arc

8 Naming Arcs G For instance, m GF = mghf = 60. m GF is read the measure of arc GF. You can write the measure of an arc next to the arc. The measure of a semicircle is always 180. E E H F

9 Naming Arcs G The measure of a GF major arc is defined as the difference between 360 and the measure of its associated minor arc. For example, m = = 300. The measure of the whole circle is 360. GEF E E H F

10 Ex. 1: Finding Measures of Arcs Find the measure of each arc of R. a. b. c. MNMPN PMN N 80 R P M

11 Ex. 1: Finding Measures of Arcs Find the measure of each arc of R. a. b. c. MNMPN PMN Solution: MN is a minor arc, so m MN = mmrn = 80 N 80 M R P

12 Ex. 1: Finding Measures of Arcs Find the measure of each arc of R. a. b. c. Solution: MPN MNMPN PMN MPN is a major arc, so m = = 280 N 80 M R P

13 Ex. 1: Finding Measures of Arcs Find the measure of each arc of R. a. b. c. PMN MNMPN PMN Solution: is a semicircle, so m = 180 PMN N 80 M R P

14 Note: C A Two arcs of the same circle are adjacent if they intersect at exactly one point. You can add the measures of adjacent areas. Postulate 26 Arc Addition Postulate. The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs. m = m AB + m ABC B BC

15 Ex. 2: Finding Measures of Arcs Find the measure of each arc. a. GE b. GEF c. GF m = m + m = GE GH HE = 120 F G R 110 H E

16 Ex. 2: Finding Measures of Arcs Find the measure of each arc. a. GE b. GEF c. m GEF = m GE + m = = 230 GF EF 40 F G 80 R 110 H E

17 Ex. 2: Finding Measures of Arcs Find the measure of each arc. a. GE 40 b. GEF R c. GF 110 m GF = m GEF = F = 130 G 80 H E

18 Ex. 3: Identifying Congruent Arcs Find the measures of the blue arcs. Are the arcs congruent? B A D C

19 Ex. 3: Identifying Congruent Arcs Find the measures of the blue arcs. Are the arcs congruent? AB B A D and DC are in the same circle and m AB = m = 45. So, AB DC DC C

20 Ex. 3: Identifying Congruent Arcs Find the measures of the blue arcs. Are the arcs congruent? P 80 Q R 80 S

21 Ex. 3: Identifying Congruent Arcs Find the measures of the blue arcs. Are the arcs congruent? PQ RS 80 Q and are in congruent circles and m PQ = m = 80. So, PQ RS RS P R 80 S

22 Ex. 3: Identifying Congruent Arcs Find the measures of the blue arcs. Are the arcs congruent? 65 X Z Y W

23 Ex. 3: Identifying Congruent Arcs Find the measures of the blue arcs. Are the arcs congruent? X Z m XY = m ZW = 65, but XY and ZW are not arcs of the same circle or of congruent circles, so XY and ZW are NOT congruent. 65 Y W

24 Using Chords of Circles A point Y is called the midpoint of if XY YZ. Any line, segment, or ray that contains Y bisects XYZ.

25 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. A AB BC if and only if B C AB BC

26 Theorem 10.5 If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc. E F DE EF, G DG GF D

27 Theorem 10.5 If one chord is a perpendicular bisector of another chord, then the first chord is a diameter. J M JK is a diameter of the circle. L K

28 Ex. 4: Using Theorem 10.4 You can use Theorem 10.4 to find m AD. 2x (x + 40) C A B

29 Ex. 4: Using Theorem 10.4 You can use Theorem 10.4 to find m AD. 2x (x + 40) C Because AD DC, and AD DC. So, m AD = m DC A B 2x = x + 40 x = 40 Substitute Subtract x from each side.

30 Theorem 10.7 In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center. AB CD if and only if EF EG. C A E F G D B

31 Exit Quiz 5 Points Find the area of the sector shown below. P 4 ft. C A Sector CPD intercepts an arc whose measure is 85. The radius is 4 ft. m = r D

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