whth52

=Chapter 9=

=9.1 Chords and arcs=

Objective- Define a circle and its associated parts, and use them in constructions. Objective- Define and use the degree measure of arcs. Objective- Define and use the length measure of arcs. Objective- Prove a theorem about chords and their intercepted arcs.

Definitions
Circle- a set of points that are equally away from each other in the same plane. Radius- a segment that extends outward from the middle/center of a circle to a point on the edge of the circle. Chord- a segment that has its endpoints on the end of the circle. Diameter- a segment that is 2 radii long and that extends from one point on the circle to another, and that goes through the center. Arc- a curved segment of a circle that is not broken. Endpoints- the end part of a segment, it can also be the starting point of a segment. Semi-circle- an arc that is half the circle, and its endpoints are the endpoints of the diameter of the circle. Minor arc- a arc that is less then the semi-circle of a circle. Major arc- a arc that is bigger then the semi-circle of a circle. Central angle- an angle made from the center of the circle by two rays. Intercepted arc- a arc that has endpoint that are on the sides of an incribed angle. Degree measure of arcs- the measure of a minor arc is the measure of the central angle of a circle. The measure of a major arc is 360 degrees minus the measure of the arcs central angle. Arc Length- The measure of an arc of a circle that is in terms of units, like centimeters, meters, etc.

Arc Length
L=M / 360° (2πr) Where M is the degree measure of an arc of a circle, L is the length, and r is the radius.

Example 1 Measure of angle S is 48°, the radius is 6. What is the arc length?

Example 2 Length is 40, radius is 10. What is the angle measure?

9.1answers

Chords and Arcs Theorems
In a circle, or in congruent circles, the arcs of congruent chords are congruent. =9.2 Tangents to circles= Objective- Define tangents and secants of circle. Objective- Understand the relationship between tangents and certain radii of circles. Objective- Understand geometry of a radius perpendicular to a chord of a circle.

Definitions
Secant- a line that intersects a circle at any two points on the circle. Tangent- a line that is in the same plane as a circle, and intersects the circle at one point and only one point. Point of Tangency- The point that a tangent intersects a circle.

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

Radius and Chord Theorem
A radius that is perpendicular to a chord of a circle bisects the chord, and cuts the chord into two equal parts.

Converse of tangent theorem
If a line is perpendicular to a radius of a circle at its endpoint on the circle, then the line is tangent to the cirlce. =9.3 Inscribed angles and arcs=

Definitions
Inscribed angle- Any angle that has a vertex that is on a circle and whose sides are chords of that circle. Angle ABC is an inscribed angle.

Inscribed angle theorem
The measure of an angle inscribed in a circle is equal to one-half the measure of the intercepted arc.

Right-angle corollary
If an incribed angle inercepts a semi-cirlce, then the angle is a right angle.

Arc-Intercept corollary
If two incribed angles intercept the same arc, then they have the same measure.

[|Inscribed angle theorems] =9.4 Angles formed by secants and tangents=

Case 1: Vertex is on circle.
A) B)

Case 3: Vertex in outside the circle.
A) B) C)

Theorem 1
If a tangent and a secant (or chord) intersect on a circle at the point of tangency, then the measure of the angle formed is one-half the measure of its intercepted arc.

Theorem 2
The measure of an angle formed by two secants or chords that intersect in the interior of a circle is one-half the sum of the measures of the arcs intercepted by the angle and its vertical angle.

Theorem 3
The measure of an angle formed by two secants that intersect in the exterior of a circle is one-half the difference of the measures of the intercepted arcs.

Example 1 angle abc is the central angle. the measure of angle abc is 110. What is the measure of arc AC? What is the measure of angle ADC? Example 2 The measure of arc EG is 80°. The measure of arc IH is 60°. What is the measure of angle EFG?

What is the measure of angle IFH? Example 3 The measure of arc JM is 100°. The measure of arc OL is 45°.

What is the measure of angle JKM? 9.4 example answers =9.5 Segments of Tangents, Secants, and Chords= Objective- Define special cases of segments related to circles, including secant-secant, secant-tangent, and chord-chord segments. Objective- Develop and use theorems about measures of the segments. (a picture i made) Tangent segment Segment AB Secant segment Segment BD External secant segment Segment BC Chord Segment DC

Theorem 1
If two lines are tangent to a circle from the same vertex, then the segments are equal. (tangent = tangent)

Theorem 2
If two secants intersect outside of a circle at a single point, the the product of the whole secant and its segment outside the circle are equal to the product of the length of the other secant is its outside segment. (Whole • outside = Whole • outside)

Theorem 3
If a secant and a tangent intersect at a single point outside a circle, the the product of the whole secants and its outside segment are equal to the tangent segment squared. (Whole • outside = Tangent squared)

Theorem 4
If two chords intersect inside a circle, the the product of one part of one chord times the other part of the same chord is equal to the product of one part of the other chord times the other part of that chord. (part 1 • Part 2 = part 1 • part 2)

=9.6 Circles In The Coordinate Plane= The equation for this type of circle is x²+y² = r ² The center of this circle is the point (0,0). The radius of this is 5. So,the equation for this cirlce is x²+y² = 25. The equation for this type of circle is (x-h)²+(y-k)² = r² The center of the circle is the point (4,5). The radius is 3. So, the equation for this cirlce is (x-4)²+(y-5)² = 9.

Some interesting sites on circles.
http://www.cobb.k12.ga.us/~smitha/CIRCLES.html http://home.byu.net/ser47/GroupPage.html A real life example of how circles are used is by farmers in crop circles. Here is a site that talks about crop circles: http://www.lovely.clara.net/crop_circles_sound.html