deas531

=Chapter 9=

Chords and Arcs
Below blue- tangents red- chords green- radius

center of the circle a major arc is 360˚ minus the degree measure of its central angle is given by L= M/360˚ * (2πr)
 * Circle**- a set of all the points in a plane that are equidistant from a given point in the plane; this is known as the
 * Chord**- a segment whose endpoints lie on a circle
 * Diameter**- a chord that contains the center of the circle
 * Radius**- double the length of a diameter; always branches out from the center of the circle
 * Arc**- an unbroken part of a circle
 * Endpoints**- two distinct points that divide a circle into two arcs
 * Semicircle**- an arc whose endpoints are endpoints of a diameter; half-circle; 180˚
 * Minor arc**- an arc shorter than a semicircle
 * Major arc**- an arc that is longer than a semicircle
 * Central angle**- an angle whose vertex is the center of the circle
 * Intercepted arc**- an arc whose endpoints lie on the sides of a central angle
 * Degree measure of arcs**- the measure of a minor arc is the measure of its central angle; the degree measure of
 * Arc Length**- if r is the radius of a circle and M is the degree measure of the circle, then the length, L, of the arc
 * Chords and Arcs Theorem**- In a circle, or in congruent circles, the arcs of congruent chords are congruent

__Example__: Find the measures of arcs AB, BC, ABC and AC in circle X.

Central angles have the same arc degree measure. So, arc AB= 130˚ and arc BC= 80˚ Arc ABC can be found by adding arc AB and arc BC together, so arc ABC= 210˚ Arc AC can be found by subtracting arc AB and arc BC, so arc AC= 150˚

Site with the theorems, definitions, and a picture: http://library.thinkquest.org/10030/13arcsandc.htm

Tangents to Circles
2 points of intersection ||||||||||||||||| 1 point of intersection |||||||||||||| No points of intersection

drawn to the point of tangency then the line is tangent to the circle
 * Secant**- a line that intersects a circle at 2 points
 * Tangent**- a line that intersects a circle at exactly one point; that point is alos called the point of tangency
 * Tangent Theorem**- If a line is tangent to a circle, then then the line is perpendicular to a radius of the circle
 * Radius and Chord Theorem**- A radius that is perpendicular to a chord of a circle bisects the chord
 * Converse of the Tangent Theorem**- If a line is perpendicular to a radius of a circle at its endpoint on the circle,
 * Theorem**- The perpendicular bisector of a chord passes through the center of the circle

__Example__: Circle L has a radius of 10 cm. and LX is 6 cm. Segment LV is perpendicular to segment AB at point X. Find the length of AB. Using the Pythagorean Theorem: (AX)² + 6²= 10² (AX)²= 10² - 6² (AX)²= 36 sqrt (AX)= 6 By the Radius and Chord Theorem, segment LV bisects segment AB, so BX= AX= 6. So AB= AX + BX= 6 + 6= 12

All about tangents, with pictures: [|http://www.geom.uiuc.edu/~dwiggins/conj42.html]



Inscribed Angles and Arcs
inscribed arc
 * Inscribed angle**- an angle with the vertex inside the circle, and 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
 * Right-angle Corollary**- If an inscribed angle intercepts a semicircle, then the angle is a right angle
 * Arc-Intercept Corollary**- If two inscribed angle intercept the same arc, then they have the same measure

__Example__: Find the angle measure of angle B. By the Arc-Intercept Corollary, angle A and angle B have the same measure, so angle A= angle B= 45˚

link with great pictures about inscribed angles, and an activity: http://www.ies.co.jp/math/java/geo/enshukaku/enshukaku.html



Angles Formed by Secants and Tangents
Case 1: Vertex is on the circle. Case 2: Vertex is in the circle. Case 3: Vertex is outside the circle. the the angle formed is half the measure of its intercepted arc half the sum of the measure of the arcs intercepted by the angle and its vertical angle difference of the measures of the intercepted arcs measures of the intercepted arcs measures of the intercepted arcs, or the measure of the major arc minus 180⁰
 * Theorem**- If a tangent and a secant (or chord) intersect on a circle at the point of tangency, then the measure of
 * Theorem**- The measure of an angle formed by two secants or chords that intersect in the interior of a circle is
 * Theorem**- The measure of an angle formed by two secants that intersect in the exterior of a circle is half the
 * Theorem**- The measure of a secant-tangent angle with its vertex outside the circle is half the difference of the
 * Theorem**- The measure of a tangent-tangent angle with its vertex outside the circle is half the difference of the

__Example__: Find the measure of angle AQC. Two secants intersect the inside circle. The the theorem that deals with secants intersecting inside circles, the measure of angle AQC is half the sum of the two arcs. So, with arc AC being 30˚, and arc BD being 70˚, the sum being 100˚, and if you divide that in 1/2, angle is 50˚.





Site with the theorems, examples, and pictures: http://regentsprep.org/regents/mathb/5A1/CircleAngles.htm

Segments of Tangents, Secants, and Chords
same length external segment equals the product of the lengths of the other secant segment and its external segment Equation: w*o=w*o its external segment equals the length of the tangent Equation: w*o=t²
 * Theorem**- If two segments are tangent to a circle from the same external point, then the segments are the
 * Theorem**- If two secants intersect outside a circle, then the product of the lengths of one secant segment and its
 * Theorem**- If a secant and a tangent intersect outside a circle, then the product of the lengths of the segment and

__Example__: AX= 9, XC= 5, DX= 7. What is XB?. Segments AX, XC, DX, and XB are intersecting chords in a circle. By the theorem dealing with chords that intersect inside a circle: pt1 * pt2= pt1 * pt2

AX * XC= DX * XB 9 * 5= 7x 45= 7x /7... /7 = 6.42857142857 or 6.43



Site about rules when dealing with segments that cut, touch and intersect in circles: http://www.regentsprep.org/Regents/math/geometry/GP14/CircleSegments.htm



Circles in the Coordinate Plane
Equation for a circle with the center on the origin: x² + y²= r² (r is for radius)

__Example__: What is the equation for the circle below?



You already know that when x or y don't change, they are 0. In this case, both of them are 0, so you the center is on the origin. Now all you have to do is worry about the radius, and that's 6. x² + y²= 36

Equation for a circle with the center NOT on the origin: (x-h)² + (y-k)² = r²

__Example__: What is the equation for the circle below?



First, you need to know the center and radius. center= -3, 3 radius= 3 So the equation is (x+3)² + (y-3)²= 9

Site with pictures explaining both equations: http://regentsprep.org/regents/math/conics/LCir.htm