lesson 10.2 arcs and chords. arcs of circles central angle-angle whose vertex is the center of the...

Lesson 10.2 Arcs and Chords

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Minor Arc formed from a central angle less than 180° minor arc

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Lesson 10.2

Arcs and Chords

Arcs of Circles

• Central Angle-angle whose vertex is the center of the circle.

central angle

Minor Arc

• formed from a central angle less than 180°

minor arc

Page 4: Lesson 10.2 Arcs and Chords. Arcs of Circles Central Angle-angle whose vertex is the center of the circle. central angle

Major Arc

• formed from a central angle that measures between 180 ° - 360 °

major arc

Page 5: Lesson 10.2 Arcs and Chords. Arcs of Circles Central Angle-angle whose vertex is the center of the circle. central angle

Semicircle

• formed from an arc of 180 °• Half circle!• Endpoints of an arc are endpoints

of the diameter

Page 6: Lesson 10.2 Arcs and Chords. Arcs of Circles Central Angle-angle whose vertex is the center of the circle. central angle

Naming Arcs

• How do we name minor arcs, major arcs, and semicircles??

Page 7: Lesson 10.2 Arcs and Chords. Arcs of Circles Central Angle-angle whose vertex is the center of the circle. central angle

Minor Arc

• Named by the endpoints of the arc.

Minor Arc: AB or BA

Page 8: Lesson 10.2 Arcs and Chords. Arcs of Circles Central Angle-angle whose vertex is the center of the circle. central angle

Major Arc

• Named by the endpoints of the arc and one point in between the arc

Major Arc: ACB or BCA

Could we name this major arc BAC?

Page 9: Lesson 10.2 Arcs and Chords. Arcs of Circles Central Angle-angle whose vertex is the center of the circle. central angle

Semicircle

• Named by the endpoints of the diameter and one point in between the arc

mABC = 180°

Page 10: Lesson 10.2 Arcs and Chords. Arcs of Circles Central Angle-angle whose vertex is the center of the circle. central angle

Example

Page 11: Lesson 10.2 Arcs and Chords. Arcs of Circles Central Angle-angle whose vertex is the center of the circle. central angle

Measuring Arcs• A Circle measures 360 °

Page 12: Lesson 10.2 Arcs and Chords. Arcs of Circles Central Angle-angle whose vertex is the center of the circle. central angle

Measure of a Minor Arc

• Measure of its central angle

95°m AB=95 °

Page 13: Lesson 10.2 Arcs and Chords. Arcs of Circles Central Angle-angle whose vertex is the center of the circle. central angle

Measure of a Major Arc

• difference between 360° and measure of minor arc

95°mACB=360°– 95° = 265°

Page 14: Lesson 10.2 Arcs and Chords. Arcs of Circles Central Angle-angle whose vertex is the center of the circle. central angle

Arc Addition Postulate

• Measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs.

45A C

What is the measure of BD?

m BD=100 °

Page 15: Lesson 10.2 Arcs and Chords. Arcs of Circles Central Angle-angle whose vertex is the center of the circle. central angle

Example for #1-10

Page 16: Lesson 10.2 Arcs and Chords. Arcs of Circles Central Angle-angle whose vertex is the center of the circle. central angle

Congruent Arcs

• Two arcs of the same circle or congruent circles are congruent arcs if they have the same measure.

AB is congruent to DC since their arc measures are the same.

Page 17: Lesson 10.2 Arcs and Chords. Arcs of Circles Central Angle-angle whose vertex is the center of the circle. central angle

Theorem 10.4

• Two minor arcs are congruent iff their corresponding chords are congruent.

Chords are congruent

Page 18: Lesson 10.2 Arcs and Chords. Arcs of Circles Central Angle-angle whose vertex is the center of the circle. central angle

Example 1Solve for x

2x X+40

Page 19: Lesson 10.2 Arcs and Chords. Arcs of Circles Central Angle-angle whose vertex is the center of the circle. central angle

Theorem 10.5

• If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc.

If DE = EF, then DG = GF

Page 20: Lesson 10.2 Arcs and Chords. Arcs of Circles Central Angle-angle whose vertex is the center of the circle. central angle

Example. Find DC.

A ED

m DC = 40º

Page 21: Lesson 10.2 Arcs and Chords. Arcs of Circles Central Angle-angle whose vertex is the center of the circle. central angle

Theorem 10.6

• If one chord is a perpendicular bisector to another chord, then the first chord is a diameter.

Since AB is perpendicular to CD, CD is the diameter.

Page 22: Lesson 10.2 Arcs and Chords. Arcs of Circles Central Angle-angle whose vertex is the center of the circle. central angle

Example. Solve for x.

7 x = 7

Page 23: Lesson 10.2 Arcs and Chords. Arcs of Circles Central Angle-angle whose vertex is the center of the circle. central angle

Theorem 10.7

• Two chords are congruent iff they are equidistant from the center.

Congruent Chords

Page 24: Lesson 10.2 Arcs and Chords. Arcs of Circles Central Angle-angle whose vertex is the center of the circle. central angle

Example. Solve for x.

x x = 15

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