- 132 Views
- Uploaded on

Download Presentation
## PowerPoint Slideshow about ' 10.2 Arcs and Chords' - gary-norris

**An Image/Link below is provided (as is) to download presentation**

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript

Objectives/Assignment

- Use properties of arcs of circles, as applied.
- Use properties of chords of circles.
- Assignment: pp. 607-608 #3-47
- Reminder Quiz after 10.3 and 10.5

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

Using Arcs of Circlesin the interior of APB form a is 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.

Using Arcs of CirclesNaming Arcs is a

- Arcs are named by their endpoints. For example, the minor arc associated with APB above is . Major arcs and semicircles are named by their endpoints and by a point on the arc.

60°

60°

180°

Naming Arcs is a

- For example, the major arc associated with APB is .
here on the right is a semicircle.

The measure of a minor arc is defined to be the measure of its central angle.

60°

60°

180°

Naming Arcs is a

- For instance, m = mGHF = 60°.
- m 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°.

60°

60°

180°

Naming Arcs is a

- The measure of a major arc is defined as the difference between 360° and the measure of its associated minor arc. For example, m = 360° - 60° = 300°. The measure of the whole circle is 360°.

60°

60°

180°

Ex. 1: Finding Measures of Arcs is a

- Find the measure of each arc of R.
Solution:

is a minor arc, so m = mMRN = 80°

80°

Ex. 1: Finding Measures of Arcs is a

- Find the measure of each arc of R.
Solution:

is a major arc, so m = 360° – 80° = 280°

80°

Ex. 1: Finding Measures of Arcs is a

- Find the measure of each arc of R.
Solution:

is a semicircle, so m = 180°

80°

Note: is 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 + m

Ex. 2: Finding Measures of Arcs is a

- Find the measure of each arc.
m = m + m =

40° + 80° = 120°

40°

80°

110°

Ex. 2: Finding Measures of Arcs is a

- Find the measure of each arc.
m = m + m =

120° + 110° = 230°

40°

80°

110°

Ex. 2: Finding Measures of Arcs is a

- Find the measure of each arc.
m = 360° - m =

360° - 230° = 130°

40°

80°

110°

- and are in the same circle and is a
m = m = 45°. So,

- Find the measures of the blue arcs. Are the arcs congruent?

45°

45°

- and are in congruent circles and is a
m = m = 80°. So,

- Find the measures of the blue arcs. Are the arcs congruent?

80°

80°

Ex. 3: Identifying Congruent Arcs is a

- Find the measures of the blue arcs. Are the arcs congruent?

65°

m = m = 65°, but and are not arcs of the same circle or of congruent circles, so and are NOT congruent.

Using Chords of Circles is a

- A point Y is called the midpoint of if . Any line, segment, or ray that contains Y bisects .

is a

if and only if

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.

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

,

Theorem 10.5If one chord is a perpendicular bisector of another chord, then the first chord is a diameter.

Theorem 10.5is a diameter of the circle.

- Because AD DC, and . So, then the first chord is a diameter.m = m

(x + 40)°

- You can use Theorem 10.4 to find m .

2x°

2x = x + 40

Substitute

Subtract x from each side.

x = 40

Theorem 10.6 can be used to locate a circle’s center as shown in the next few slides.

Step 1: Draw any two chords that are not parallel to each other.

Ex. 5: Finding the Center of a CircleStep 2: Draw the perpendicular bisector of each chord. These are the diameters.

Ex. 5: Finding the Center of a CircleStep 3: The perpendicular bisectors intersect at the circle’s center.

Ex. 5: Finding the Center of a CircleMasonry Hammer circle’s center.. A masonry hammer has a hammer on one end and a curved pick on the other. The pick works best if you swing it along a circular curve that matches the shape of the pick. Find the center of the circular swing.

Ex. 6: Using Properties of ChordsDraw a segment AB, from the top of the masonry hammer to the end of the pick. Find the midpoint C, and draw perpendicular bisector CD. Find the intersection of CD with the line formed by the handle. So, the center of the swing lies at E.

Ex. 6: Using Properties of ChordsIn 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.

Theorem 10.7AB = 8; DE = 8, and CD = 5. Find CF. congruent if and only if they are equidistant from the center.

Ex. 7: Using Theorem 10.7Ex. 7: Using Theorem 10.7 congruent if and only if they are equidistant from the center.

Because AB and DE are congruent chords, they are equidistant from the center. So CF CG. To find CG, first find DG.

CG DE, so CG bisects DE. Because DE = 8, DG = =4.

Ex. 7: Using Theorem 10.7 congruent if and only if they are equidistant from the center.

Then use DG to find CG. DG = 4 and CD = 5, so ∆CGD is a 3-4-5 right triangle. So CG = 3. Finally, use CG to find CF. Because CF CG, CF = CG = 3

Reminders: congruent if and only if they are equidistant from the center.

- Quiz after 10.3
- Last day to check Vocabulary from Chapter 10 and Postulates/Theorems from Chapter 10.

Download Presentation

Connecting to Server..