paje614

=Circles=

Chords and arcs
circle: a set of all the points in a plane that are all equal distance from a given point in the plane known as the center of the circle. A circle has 3 parts Radius- this is a segment that starts at a point in the center of the circle and end at a point on the outer edge of the circle. Chords- a segment whose endpoints line on a circle. Diameter- a chord that contains the center of a circle.

Central Angle: An angle in the plane of a circle whose vertex is the center of the circle. Intercepted arc: This is an arcs whose endpoints lie one sides of the angle and whose other points lie in the interior of the angle.

Degree measure of arcs: The central angle is the measure of a minor arc. The degree measure of a major arc is 360 degrees minus the degree measure of it's minor arc. The degree measure of a semicircle is 180 degrees, half of 360.

Finding the length of an arc: r is radius of the circle and M is the degree measure of an arc of the circle, the length,L, of the arc is given by the following::: L= M/360 degrees(2Pi r)

=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 geometry of a radius perpendicular to a chord of a circle.

Secants: A line that goes through a circle Tangent: A line that is in a plane of a circle that intersects the circle at exactly one point. Below is an example of a tangent line.

Below the hands on the clock are a good example of a secant line of 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 is perpendicular to a chord of a circle bisect 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 to the circle. Theorem: The perpendicular bisector of a chord passes through the center of the circle. =Inscribed Angles and Arcs=


 * Define inscribed and intercepted arc.
 * Develop and use the inscribed angle theorem and its corollaries.

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 inscribed angle intercepts a semicircle, then the angle a a right angle. Arc-Intercept Corollary: If two inscribed angles intercept the same arc, then they have the same measure.

=Angles Formed 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.

Theorem 1: If a tangent and a secant(or a 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. Theorem 4: The measure of a secant-tangent angle with its vertex outside the circle is one half the difference of the measures of the intercepted arcs. Theorem 5: The measure of a tangent-tangent angle with its vertex outside the circle is one half the difference of the intercepted arcs, or the measure of the major arc minus 180 degrees.

=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.

Theorem 1: If two segments are tangent to a circle from the same external point, then the segments are equal. Theorem 2: If two secants intersect outside a circle, the product of the lengths of one secant and its external segment equals the product of the lengths of the other secant segment and its external segment. (Remember: Whole*Outside=Whole*Outside) Theorem 3: If a secant and a tangent intersect outside a circle, then the product of the lengths of the segment and its external segment equals the length of the tangent segment squared. (Remember: Whole*Outside=Tangent squared) Theorem 4: If two chords intersect inside a circle, then the product of the lengths of the segments of one chord equals the product of the lengths of the segments of the other chord.

=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.

When the origin is at (0,0) use this equation X squared+Y squared=Radius squared

For example When X squared+Y squared=25 The radius=5 The center=(0,0)

Here try this one! Activity 1 When X squared+Y squared= 100 The radius would be...? The center would be...?

What about when the center of the circle isn't at the origin but at a NEW point? WELL.. you use the point called (h,k)

Use this equation! (X-h)squared+(Y-k)squared=Radius Squared

For example! (X-5)squared+(Y-4)squared=3squared The center=(5,4) The radius=9

Try this one! Activity 2 (X-10)squared+(Y+2)squared=2squared..

Can you tell me what the radius is..? what about the center..?

Answers for examples 1&2 on lesson 9.6...yeah