ircles
Tangents to ircles circle is the set of all points in a plane that are equidistant from a given point called the center of the circle. circle with center P is called circle P. The distance from the center to a point on the circle is the radius of the circle. Two circles are congruent if they have the same radius. The distance across the circle, through its center, is the diameter of the circle. The diameter is twice the radius. radius diameter center The term radius and diameter describe segments as well as measures. radius is a segment whose endpoints are the center of the circle and a point on the circle. ll radii of circles are congruent.
Q R k P S chord is a segment whose secant is a line that intersects Endpoints are points on the circle. circle in two points. Line j is a secant. PS andpr arechords. diameter is a chord that passes tangent is a line in the plane of a through the center of the circle. ircle that intersects the circle in PR isadiameter. exactly one point. Line k is a tangent j
In a plane, two circles can intersect in two points, one point, or no points. oplanar circles that intersect in one point are called tangent circles. oplanar circles that have a common center are called concentric. points of intersechon point of intersechon (tangent circles) No point of intersechon oncentric circles line or segment that is tangent to two coplanar circles is called a common tangent. common internal tangent intersects the segment that joins the centers of the two circles. common external tangent does not intersect the segment that joins the centers of the two circles.
The point at which a tangent line intersects the circle to which it is tangent is the point of tangency. Theorem: If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency. If l is tangent to ʘ Q at P, then l QP Q P l Theorem: In a plane, if a line it perpendicular to a radius of a circle at its endpoint on the circle, then the line is tangent to the circle. P If l QP, then l is tangent to ʘ Q at P Q l
Theorem: If two segments from the same exterior point are tangent to a circle, then they are congruent. If SR and ST are tangent to ʘP, then ST SR P T R S
rcs and hords In a plane, an angle whose vertex is the center of the circle is a central angle of the circle. If the measure of a central angle, PB is less than 80, then and B and the points of ʘP in the interior of PB form a minor arc of the circle. The points and B and the points of ʘP in the exterior of PB form a major arc of the circle. If the endpoints of an arc are the endpoints of a diameter, then the arc is a semicircle. Major arc entral angle P B Minor arc
Naming rcs: rcs are named by their endpoints. For example, the minor arc associated with PB on the previous slide is B. Major arcs and semicircles are named by their endpoints and by a point on the arc. For example, the major arc associated with PB on the previous slide is B. Measuring rcs: The measure of a minor arc is defined to be the measure of its central angle. For example, mgf = m GHF = 60 The measure of a major arc is defined as the difference between 360 and the measure of its associated minor arc. For example, mgef = 360-60 = 300 E H 80 60 G 60 F
rc ddition Postulate: The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs. mb = mb + mb B Theorems bout hords of ircles In the same circle, or in congruent circles, two minor arcs are congruent is and only if their corresponding chords are congruent. B B if andonly if B B B If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc. DE EF, DG GF If one chord is a perpendicular bisector of another chord, then the first chord is a diameter. JK isadiameter of the circle J E G D F M K L
Theorem: In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center. B D if andonly if EF EG E G F B D
Inscribed ngles n inscribed angle is an angle whose vertex is on a circle and whose sides contain chords of the circle. The arc that lies in the interior of and inscribed angle and has endpoints on the angle is called the intercepted arc of the angle. Inscribed angle Intercepted arc Theorem: If an angle is inscribed in a circle, then its measure is half the measure of its intercepted arc. m DB = mb D B
Theorem: If two inscribed angles of a circle intercept the same arc, then the angles are congruent. B D D If all the vertices of a polygon lie on a circle, the polygon is inscribed in the circle, and the circle is circumscribed about the polygon. Theorem: If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle. onversely, if on side of an inscribed triangle is a diameter of the circle, then the triangle is a right triangle and the angle opposite the diameter is the right angle B is a right angle if and only if is a diameter of the circle. B
Theorem: quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary. D, E, F, and G, lie on some circle, ʘ, if and only if m D + m F = 80 and m E + m G = 80 E D F G
Other ngle RelaHonships in ircles Theorem: If a tangent and a chord intersect at a point on a circle, then the measure of each angel formed is one half the measure of its intercepted arc. B m = mb m = mb
Theorem: If two chords intersect in the interior of a circle, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle. m = + ( md + mb),m = ( mb md) B D Theorem: If a tangent and a secant, two tangents, or two secants intersect in the exterior of a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs. P B X W 3 Q Y Z R = ( mb m ) m = ( mpqr mpr ) m 3 = ( mxy mwz ) m
Segment Lengths in ircles When two chords intersect in the interior of a circle, each chord is divided into two segments which are called segments of a chord. The following theorem gives a relationship between the lengths of the four segments that are formed. Theorem: If two chords intersect in the interior of a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord. B E EB = E ED E D
In the figure shown below, PS, is called a tangent segment because it is tangent to the circle at an endpoint. Similarly, PR is a secant segment and PQ is the external segment of PR External secant segment P Q R Tangent segment S
Theorem: If two secant segments share the same the same endpoint outside a circle, then the product of the length of one secant segment and the length of its external segment equals the product of the length of the other secant segment and the length of its external segment. E EB = E ED B E D Theorem: If a secant segment and a tangent segment share an endpoint outside a circle, then the product of the length of the secant segment and the length of if its external segment equals the square of the length of the tangent segment. ( E) = E ED E D
EquaHons of ircles You can write an equation of a circle in the coordinate plane of you know its radius and the coordinates of its center. Suppose the radius of a circle is r and the center is (h, k). Let (x, y) be any point on the circle. The distance between (x, y) and (h, k) is r, so you can use the Distance Formula. ( x h) + ( y k) = r y r (h, k) (x, y) x Square both sides to find the standard equation of a circle with radius r and center (h, k). Standard equation of a circle: ( ) ( ) x h + y k = r If the center is the origin, then the standard equation is x + y = r
Locus Example: Draw point on a piece of paper. Draw and describe the locus of all locus points in on a plane the paper is the that set are of all 3 points inches in from a plane. that satisfy a given condition or a set of given conditions. The word locus is derived from the Latin word for location. The plural of locus is loci. locus is often described as the path of an object moving in a plane. For instance, the reason that many clock faces are circular is that the locus of the end of a clock s minute hand is a circle. Finding a Locus To find the locus of points that satisfy a given condition, use the following steps.. Draw any figures that are given in the statement of the problem.. Locate several points that satisfy the given condition. 3. ontinue drawing points until you can recognize the pattern. 4. Draw the locus and describe it in words.
Example : Points and B lie in a plane. What is the locus of points in the plane that are equidistant from points and B and are a distance of B from B? Example 3: Point P is in the interior of B. What is the locus of points in the interior of B that are equidistant from both sides of B and inches from P? How does the location of P within B affect the locus.