perjor211

CHAPTER NINE -circles
=9.1 chords and arcs.=

= = =circle= =a circle is the set of all points in a plane that are equidistant from a given point i n 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 end points line on a circle=

=diameter= =is a chord that contains the center of a circle=



the degree measure if a major arc is 360^ minus the degree measureof its minnor arc.the degree measure of a semicircle is 180
^



find the lenght of the indicated arc.express your answer to the nearest millimeter.(there are 20 equal sectors on a dart board.
==SOLUTION the lenght of an arc is 1/20 of the circumference of the circle.remember that c=2 pi r ==

=17 pi= 53.4=53 mm =17 pi= 53.4=53 mm

==9.2 secants and tangents a secant to a circle is the line that intersects the circle at two points. a tangent is a line in the plane of the circle that intersects the circle at exactly one point, wich is known as the point of tagency.==

==tangent theorem if the line is a tangent to the circle then the line is end to a radius of the circle drawn to the point of tangency radius and chord theorem. a radius that is perpendicular to a chord of the circle intersects 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 circle is radii perpendicular to chords theorem. the perpendicular bisector of a chord passes throught the center of the circle==

an inscribed angle is 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 interior 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 corollary if two inscribed angles intercept the same arc, then they have the same measure

9.4 angles formed by secants and tangents

angles formed by a pair of lines that intersects a circle in two or more places can be study systematically.there are three cases to consider, according to the placement of the vertex of the angles. every man is a vertex on the circle vertex inside the circle vertex outside the angle theorem if a tangent and a secant intesct on a circle at the poin of tangency, then the measure of the angle formed is vertex measure of its intercepted arc

theorem the measure of an angle formed by two secants or chords that intesect in the interior of a circle is vertex the angle of the arcs intercepted by the angle and its vertical angle. 9.5 segments of tangents secants, and chords

theorem if two segments are tangent to a circle from external point, then the segments intersect theorem if two secants intersect outside a circle, the product of the lenghts of one secant segment and its external segment equals ? (whole x outside=whole x outside)

Theorem : If two chords, seg.AB and seg.CD intersect inside or outside a circle at P then l (seg. PA) ´ l (seg. PB) =l (seg. PC) ´ l (seg. PD)= = =**Theorem**= =The lengths of two tangent segments from an external point to a circle are equal.= = = = = =Example Two chords seg AB and seg. CD intersect in the circle at P. Given that l (seg.PC)= =l (seg.Pb)= =1.5 cm and l (seg.PD)= 3 cm. Find l (seg.AP).= == == =**Solution:** l (seg AP) ´ l (seg PB)= l (seg DP) ´ l (seg PC) l (seg AP) ´ 1.5 =3 ´ 1.5 l (seg AP)= 3 cm.=

=**The measure of an angle formed by two secants intersecting the exterior of the circle is one-half the difference of the measures of the intercepted arcs.**= =**If a tangent and a secant or a chord intersects at the point of tangency on a circle, then the measure of the angle formed is one-half the measure of its intercepted arc.**= =**The measure of an angle formed either by a tangent and a secant intersecting at a point exterior to a circle**= =**two tangents intersecting at a point exterior to a circle equals one-half the difference of the measures of the intercepted arcs.**= =**If two chords of a circle intersect, 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.**= =**If a tangent and a secant intersect in the exterior of a circle, then the square of the length of the tangent segment equals the product of the lengths of the secant segment and the external secant segment.**= =**If two secants intersect in the exterior of a circle, then the product of the lengths of one secant segment and its external segment equals the product of the lengths of the other secant segment and its external segment.**= = = =**When you want to represent a circle on a coordinate plane, you need to use the following equation: (x-h)^2 + (y-k)^2**= r^2, where h and k are the center points of the circle and r is the radius or the circle. u can use also the new method= =jaja=