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Mark Letizio's amazing chapter 9 page

9.1 CHORDS AND ARCS http://www.automotivebuzz.com/Images/Media/Automotive-Buzz/MediaCat44.jpg CIRCLE is a set of all points in a plane that are equidistant form a given point in the plane known as the center of the circle.

RADIUS is a segment from the center of the circle to a point on the circle.

CHORD is a segment whose endpoints line on a circle.

DIAMETER is a chord that contains the center of a circle.

SEMICIRCLE is an arc whose endpoints are endponits of a diameter.

MINOR ARC of a circle is an arc that is shorter than a semicircle of that circle.

MAJOR ARC of a circle is an arc that is longer than semicircle of that circle.

CENTRAL ANGLE of a circle is an angle in the plane of a circle whose vertex is the center of the circle.

INTERCEPTED ARC is an arc whose endponits lie on the sides of the angle and whose other points lie in the interior of the angle of the central angle.

ARC LENGTH..... L=M/360(2TTr)

http://mathworld.wolfram.com/images/eps-gif/MidArcPoints_1000.gif

9.2 TANGENTS TO CIRCLES

SECANT to a circle is a line that intersects the circle at two points.

TANGENT is a line the plane of the circle that intersects the circle at exactly one point, which is known as the point of tangency.

THEOREM the perpendicular bisector of a chord passes through the center of the 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 or tangency.

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 endpoints on the circle, then the line is tangent to the circle.

9.3 INSCRIBED ANGLES AND ARCS

INSCRIBED ANGLE is an an whose vertex lies on a circle and whose sides are chords of the circle.

RIGHT-ANGLE COROLLARY If an inscribed angle intercepts a semicircle, 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 ONE

Find the measure of XVY

Solution: XVY is inscribed in P and intercepts XY. By the Inscribed Angle Theroem: m XVY 1/2(45) 221/2

EXAMPLE TWO

A person's effective field of vision is about 30 degrees. In the diagram of the amphitheater, a person sitting at point A can see the entire stage. What is the measure of B? Can the person sitting at point B view the entire stage?

Solution: Angles A and B intercept the same arc. By Corollary 9.3.3, the angles must have the same measure, so m A= m B =30. The person sitting at point B can view the entire stage.

9.4 ANGLES FORMED BY SECANTS AND TANGENTS

THEROEM 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.

THEROEM The measure of and angle formed by two secants or chords that intersect in the interior of a circle is one-half the sum of the measure of the arcs intercepted by the angle and its verticle.

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

9.5 SEGMENTS OF TANGENTS,SECANTS, AND CHORDS

THEROEM If two segmentsare tangent to a circle from the same external point, then the segments are of equal length.

THEROEM If two secants intersect outside a circle, the the product of the lengths of one secant segment and its external segment equals the product of the lengths of the other secants segments and its external segment. (Whole x Outside = Whole x Outside)

THEROEM 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 length of the tangent segment squared.

EXAMPLE ONE Global positioning satellites are used in navigation. If the range of the satellite, AX, is 16,000 miles, what is BX?

Solution: AX and BX are tangents to a circle from the same external point. By Theroem 9.5.1, they are equal.

AX= BX =16,000 miles

9.6 CIRCLES IN THE COORDINATE PLANE EXAMPLE ONE Given: x2 + y2 = 25 Sketch and describe the graph by finding ordered pairs that satisfy the equation. Use a graphics calculator to varify your sketch.
 * http://www.youth-basketball-tips.com/images/courtdiagram.jpg**

Solution: When sketching the graph of a new type of equation, it is often helpful to locate the intercepts. To find the x-intercept, find the values of x when y = 0

x2 +0sq = 25 x2 = 25 x = (+)(-)5 Thus, the graph has two x-intercepts, (5,0) and (-5,0)