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
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)
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)
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.
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)
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 one-half the measure of its intercepted arc. Examples: Find the value of the variable. 9) 10)
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 one-half 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 one-half 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)
a(b) = c(d) w(x + w) = y(z + y) t 2 = y(z + y) Examples: Find the value of the variable.
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
Example: Graph the circles: 8) x 2 + y 2 = 16 9) (x-4) 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)
12.6: Locus Locus: set of all points in a plane that satisfies a given condition Loci plural of locus, pronounced low-sigh 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.