12.1: Tangent Lines

Congruent Circles: circles that have the same radius length

Diagram of ExamplesCenter 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:

DiagramA 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:

DiagramInscribed Angle and Intercepted Arc:

A

C B

Theorem 12.11: Inscribed Angle TheoremIf 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) t2 = 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 = r2

Center = (h, k) Radius = r

Example: Write the standard equation of a circle given1) centered at (3,15) that covers a radius of 7 2) center (2, 5); r =

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 circle6) Write the equation for a circle 7)with center (1, -3) and passing through (2, 2).You will need to use the distance formula.

x

y

Example: Graph the circles:8) x2 + y2 = 16 9) (x-4)2 + (y + 3)2 = 4

10) Write an equation of a circle with diameter . The endpoints are given.

A(0, 0), B(6, 8)

x

y

x

y

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

3) points equidistant from the endpoints of 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 .