MA_GR_5_13

=Chapter 9=

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

Definitions: Circle- the set of all points in a plane that are equidistant from a given point in the plane known as the center of the circle. Radius- a segment from the center of the circle to a point on the circle. Chord- a segment whose endpoints line on a circle. Diameter- a chord that contains the center of a circle. Arc- an unbroken part of a circle. Endpoints- the two distinct points on a circle that divide the circle into two arcs. Semicircle- an arc whose endpoints are endpoints of a diameter. Informally called a half-circle. Minor Arc- an arc that is shorter than a semicircle of a circle. Major Arc- an arc that is longer than a semicircle of a circle. Central Angle- an angle in the plane of a circle whose vertex is the center of the circle. Intercepted Arc- an arc whose endpoints lie on the sides of the angle and whose other points lie in the interior of the angle of the central angle. Degree Measure of Arcs- The degree measure of a minor arc is the measure of its central angle. The degree measure of a major arc is 360 degrees minus the degree measure of its minor arc. The degree measure of a semicircle is 180 degrees.

Blue Box Arc Length: If "r" is the radius of a circle and "M" is the degree measure of an arc of the circle, then the length, "L", of the arc is given by the following: L= M/360 degrees (2 x pi x r).

Theorems: Chords and Arcs Theorem- In a circle, or in congruent circles, the arcs of congruent chords are congruent.

The Converse of the Chords and Arcs Theorem- In a circle or in congruent circles, the chords of congruent arcs are congruent.

Example: Find the measures of arcs XZ, ZQ and XZQ. The measures of arc XZ and arc ZQ are found from their central angles. the measure of arc XZ is 100 degrees. The measure of arc ZQ is 90 degrees. Arc XZ and ZQ are adjacent angles. Now, add their measures together to find the measure of arc XZQ. Solving: The measure of arc XZQ = measure of arc XZ + ZQ. XZ + ZQ = 100 degrees + 90 degrees. Answer: 190 degrees.

9.2 Tangents to Circles
Objectives: -Define tangents and secants of circles. -Understand the relationship between tangents and certain radii of circles. -Understand the geometry of a radius perpendicular to a chord of a circle.

Definitions: Secant- A line that intersects a circle at two points. Tangent- A line in the of the circle that intersects the circle at exactly one point. Point of Tangency- The point that the lines of the circles of the tangents meet at.

Theorems: Tangent Theorem- If a line is tangent to a circle, then the line is perpendicular to the radius of a circle drawn to the point of tangency. Radius and Chord Theorem- A radius that is perpendicular to a chord of a circle is half the chord. Converse of the Tangent Theorem- If a line is perpendicular to a radius of a circle at its endpoint on the circle, then the line is tangent. Theorem- The perpendicular bisector of a chord passes through the center of the circle.

Example: Segment ry AB is perpendicular to xz DF at E, and A is the center of the circle. If AB equals 15 and AE equals 6, what is DE? Solving: AB - AE equals 9 Answer: DE equals 9

9.3 Inscribed Angles and Arcs
Objectives: -Define 'inscribed angle' and 'intercepted arc'. -Develop and use the Inscribed Angle Theorem and its corollaries.

Definitions: Inscribed Angle- An angle whose vertex lies on a circle and whose sides are chords of the circle. Intercepted Arc- The arc of a circle within an inscribed angle.

Theorems: Inscribed Angle Theorem- The measure of an angle inscribed in a circl is equal to half the measure of the intercepted arc. Right Angle Corollary- If an inscribed angle intercepts a semi-circle, then the angle is a right angle. Arc-Intercept Corollary- If two inscribed angles intercept the same arc, then they have the same measure.

Example: Find the measure of MNO. MNO is inscribed in G, and it intercepts MO. Solving: By the Inscribed Angle Theroem: Answer: Measurement of MNO 1/2(45) 221/2.

9.4 Angles Formed by Secants and Tangents
Objectives: -Define angles formed by secants and tangents of circles. -Develop and use theorems about measures of arcs intercepted by these angles.

Definitions: There are no vocabulary terms within this section that haven't already been defined above.

Theorems: Theorem Case One- If a tangent or a secant (or a chord) intersects on a circle at the point of tangency, then the measure of the angle formed is half the measure of its intercepted arc. Theorem Case Two- The measure of an angle formed by two secants or chords that intersect in the interior of a circle is blank the blank of the measures of the arcs intercepted by the angle and its vertical angle. Theorem Case Three- The measure of an angle formed by two secants that intersect in the exterior of a circle is blank the blank of the measures of the intercepted arcs.

Example: A circle is intersected by a tangent and a secant, which forms the angle BFD. The minor chord that it intersects is 180 degrees. Find the measure of angle BFD. Solving: To find the measure of BFD you divide the chord it intersects by 2. Answer: 180/2=90. ABC is 90 degrees.

9.5 Segments of Tangents, Secants, and Chords
Objectives: -Define special cases of segments related to circles, including secant-secant, secant-tangent, and chord-chord segments. -Develop and use theorems about measures of the segments.

Definitions: There are no vocabulary terms within this section that haven't already been defined above.

Theorems: Segments Formed by Tangents Theorem- If two segments are tangent to a circle from the same external point, then the two segments are equal. Segments Formed by Secants Theorem 1- If two secants intersect outside a circle, the product of the lengths of one secant segment and its external segment equals the other secant. Segments Formed by Secants Theorem 2- If a secant and a tangent intersect outside a circle, then the product of the lengths of the secant segment and its external segment equals the tangent squared. Segments Formed by Intersecting Chords Theorem- If two chords intersect inside a circle, then the product of the lengths of the segments of one chord equals the other chord.

Example: The span of a metal detector is 10 feet. IO, is 10 feet, what is JO? Solving: IO and JO are tangents to a circle from the same external point. This means that they are equal.

Answer: IO =JO= 10 feet

9.6 Circles in the Coordinate Plane
Objectives: -Develop and use the equation of a circle. -Adjust the equation for a circle to move the center in a coordinate plane.

Definitions: There are no vocabulary terms within this section that haven't already been defined above.

Theorems: There are no theorems that needed to be known for the completion of this section.

Equations: Equation 1: "To derive the equation of a circle"- To find any point that is on a circle that's not on the x- or y- axis, you can draw a triangle whose legs have lengths of x and y. The length of the hypotenuse is the distance, //r//, from the point to the origin. To find the point, use this equation: //x squared + y squared equals r squared// Equation 2: "To move the center of a circle"- To find the standard form of the equation of a circle centered at a point not on the origin, study the diagram on the Holt Geometry book on page 612. The equation for that diagram is: //(x - h squared) + (y - k squared) = r squared//

Example: The center of this circle is at the origin (0,0). What are the intercepts of the circle? Solving: Look at the common intercepts of this section. Answer: X² + Y² has the center of (0,0).

1. For a fun way to learn more about chords and arcs, try this site!: http://www.explorelearning.com/index.cfm?method=cResource.dspDetail&ResourceID=175 2. A site with a fun activity that will help you learn further of tangents to circles: http://www.netsoc.tcd.ie/~jgilbert/maths_site/applets/circles/tangents_to_circles.html 3. A site that shows several ways of spotting an inscribed angle, and a fun applet that is a 'hands on' way of making inscribed angles: http://www.ies.co.jp/math/java/geo/enshukaku/enshukaku.html 4. For a list of common formulas for angles of chords, radii, secants, and tangents, click this link. It is also full of pictures that will let you visualize the problem: http://regentsprep.org/regents/mathb/5A1/CircleAngles.htm 5. For a more detailed look into the theorems and rules of secants, tangents, and chords in circles, check out this colorful website: http://regentsprep.org/regents/mathb/5A1/CircleAngles.htm
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