Cirlces



Chords and Arcs

365845422_134a16ee16_m.jpg
365845422_134a16ee16_m.jpg


Circle - A circle is the set of all points that are equidistant from a given point in the plane known as the center of the circle.

Radius - A segment from the center of a 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 - Any two distinct point on a circle that divide the circle into two arcs.

Semicircle - An arc whose endpoints are the endpoints of a diameter.

Minor arc - An arc that is shorter than a semicircle of that circle.

Major arc - An arc that is longer than a semicircle of that 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.

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 meausre of a semicircle is 180 degrees.

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 (2PIEr)

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


Tangents to Circles

365455942_39af41f11e_m.jpg
365455942_39af41f11e_m.jpg


Seacant -
A line that intersects the circle at two points.

Tangent - A line in the plane of a circle that intersects the circle at exactly one point.

Point of Tangency - The point that the tangent intersects the circle at.

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

Radius and Chord Theorem - A radius that is perpendicular that is perpendicular to a chord of a circle opposite the chord.

Convverse 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 circle.


Inscribed Angles and Arcs

6050999_fabecc7a16_m.jpg
6050999_fabecc7a16_m.jpg


Inscribed Angle -
An angle whose vertex lies on a circle and whose sides are chords of the circle.

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

Right-Angle Corollary - If an inscribed angle intercepts a semicircle then the angle is a right angle.

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


Angles Formed by Secants and Tangents

181120439_d5c9a64d34_m.jpg
181120439_d5c9a64d34_m.jpg


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

Theorem - The measure of an angle formed by two seacants or chords that intersect in the interior of a circle is half the sum of the measures of two arcs intercepted by the angle and it's vertical angle.

Theorem - The measure of an angle formed by two seacants that intersect in the exterior of a circle.


Segments of Secants, Tangents, and Chords

260855540_19341bba05_m.jpg
260855540_19341bba05_m.jpg


Theorem -
If two segments are tangent to a circle from the same external point then the segments are perpendicular.

Theorem - If two secants intersect ouside a circle, the product of the lengths of one secant segment and its external segment equals whole times outside. (WxO=WxO)

Theorem - If a secant and a a tangent intersect outside a circle, then the product of the lengths of the secant segment and its external segment equals tangent squared. (WxO=Tangent Squared)

Theorem - If two chords intersect inside a circle, then the product of the lengths of the segments of one chord equals the length of the other chord and segment.


Circles in the Coordinate Plane

487904943_cdc9947ae5_m.jpg
487904943_cdc9947ae5_m.jpg


Objective 1
- Develop and use the equation of a circle.

Objective 2 - Adjust the equation for a circle to move the center in a coordinate plane.


Examples - http://regentsprep.org/regents/mathb/5A1/CircleAngles.htm


Helpful Websites:
-http://www.ies.co.jp/math/java/calc/limsec/limsec.html
-http://www.algebralab.org/lessons/lesson.aspx?file=Geometry_CircleSecantTangent.xml
-http://www.ima.umn.edu/~arnold/calculus/secants/secants2/secants-g.html
-http://regentsprep.org/regents/mathb/5A1/CircleAngles.htm
-http://www.mathopenref.com/secantsintersecting.html