Circle Name: Radius: Diameter: Chord: Secant:


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1 12.1: Tangent Lines Congruent Circles: circles that have the same radius length Diagram of Examples Center of Circle: Circle Name: Radius: Diameter: Chord: Secant: Tangent to A Circle: a line in the plane that intersect a circle at one exact point Point of Tangency: point at which the tangent line intersects the circle Theorem 12.1: If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency. Theorem 12.2: In a plane, 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. Theorem 12.2: If two tangent segments to the same circle come from the same point outside the circle, then the two tangent segments are congruent
2 Examples: Find the value of x in the following. 1) 2) Examples: Find the radius of each circle. 3) 4) 5) Determine the perimeter of the polygon. Assume that all lines are tangent to the circle. Examples: Determine if the line is tangent to the circle. 6) 7)
3 12.2: Chords and Arcs Central Angle: Minor Arc: Major Arc: Semicircle: Diagram A Central Angle: Minor Arc: B Major Arc: C Semicircle: D Theorem 12.4: In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent. Theorem 12.5: Central angles are congruent if and only if their chords are congruent. Theorem 12.6: Chords are congruent if and only if their arcs are congruent. Examples: Given that the circles are congruent, what can you conclude based on the figures. 1) 2) Theorem 12.7: In the same circle, or congruent circles, two chords are congruent if and only if they are equidistant from the center. Examples: Find the value of x. 3) 4)
4 Theorem 12.8: If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc. Examples: Find the value of x. 5) 6) Theorem 12.9: If a diameter bisects a chord, then it s perpendicular to the chord. Theorem 12.10: The perpendicular bisector of a chord contains the center of the circle. If you want to find the center, bisect 2 chords and find the point that they meet at.
5 12.3: Inscribed Angles Inscribed Angle: Intercepted Arc: Diagram A Inscribed Angle and Intercepted Arc: C B Theorem 12.11: Inscribed Angle Theorem If an angle is inscribed in a circle, then the measure is half the measure of the intercepted arc. Inscribed Polygon: all vertices of a polygon lie on the circle, the circle is drawn around Circumscribed: when a circle is drawn about a figure Corollary to 12.11: Two inscribed angles that intercept the same arc are congruent. Corollary to 12.11: An angle inscribed in a semicircle is a right angle. Corollary to 12.11: The opposite angles of an inscribed quadrilateral are supplementary (ADD TO 180) Examples: Find the value of each variable. 1) 2) 3)
6 4) 5) 6) 7) 8) Theorem 12.12: If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is onehalf the measure of its intercepted arc. Examples: Find the value of the variable. 9) 10)
7 12.4: Angles Measures and Segment Lengths Secant Segment: segment that extends through the circle Theorem 12.13: If two chords or secants intersect inside the circle, then the measure of each angle formed is onehalf the sum of the measures of the arcs intercepted by the angle and its vertical angle. Example: Find the value of x. Theorem 12.14: If a tangent and a secant, two tangents, or two secants intersect on the outside of the circle, the measure of the angle formed is onehalf the difference of the intercepted arcs. Examples: Find the value of x. Examples: Find the value of the numbered angle. 1) 2) 3) 4) 5) 6)
8 a(b) = c(d) w(x + w) = y(z + y) t 2 = y(z + y) Examples: Find the value of the variable.
9 12.5: Circles in the Coordinate Plane Circle: set of all points in a plane that are equidistant from a given point known as the center of the circle Equation of a Circle (x h) 2 + (y k) 2 = r 2 Center = (h, k) Radius = r Example: Write the standard equation of a circle given 1) centered at (3,15) that covers a radius of 7 2) center (2, 5); r = 2 Example: Identify the center and radius of the following circles. 3) (x 3) 2 + (y 2) 2 = 9 4) (x +2) 2 + (y + 1) 2 = 4 5) (x + 4) 2 + (y  2) 2 = 3 Example: Write the equation of the given circle 6) Write the equation for a circle 7) with center (1, 3) and passing through (2, 2). You will need to use the distance formula. y x
10 Example: Graph the circles: 8) x 2 + y 2 = 16 9) (x4) 2 + (y + 3) 2 = 4 y y x x 10) Write an equation of a circle with diameter AB. The endpoints are given. A(0, 0), B(6, 8)
11 12.6: Locus Locus: set of all points in a plane that satisfies a given condition Loci plural of locus, pronounced lowsigh Examples: Sketch each set of loci. Then describe each set. 1) points 1.5 cm from a point T 2) points 1 in. from PQ 3) points equidistant from the endpoints of AB 4) points that belong to a given angle or its interior and are equidistant from the sides of the given angle Sketching a Locus for Two Conditions Example: Sketch the locus of points that are equidistant from X and Y and 2cm from the midpoint of XY.
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