Chapter 12.1 Common Core – G.C.2 Identify and describe relationships among inscribed angels, radii, and chords…the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Objective – To use properties of a tangent to a circle.

Chapter 12.1 Notes Circles – is the set of all pts in a plane that are equidistant from a given pt, called the center

Internally and Externally Tangents The Circles are internally and externally tangent. The Lines are

Thm – If a line is tangent to a circle, it is ⊥ to the radius drawn to the point of tangency If then Thm – In a plane, if a line is ⊥ to a radius of a circle at its endpts. on the circle, then the line is tangent to the circle.

Thm – If 2 segments from the same exterior pt are tangent to a circle, then they are ≌. If then

Chapter 12.2 Notes Common Core – G.C.2 Identify and describe relationships among inscribed angles, radii, and chords. Objectives – To use congruent chords, arcs, and central angles. To use perpendicular bisectors to chords.

Chapter 12.2 Notes Central Angle – an angle made with the center of the circle. Minor Arc – is a central angle less than 180° Semicircle – is a central angle that is exactly 180° Major Arc – is a central angle more than 180°

Arc Addition Postulate m ABC = m AB + m BC Thm – AB ≌ BC AB ≌ BC Thm – DE ≌ EF, DG ≌ GF

Thm – JK is a diameter of the circle Thm – AB ≌ CD EF ≌ EG

Chapter 12.3 Notes Common Core – G.C.2, G.C.3, G.C.4 Identify and describe relationships among inscribed angles, radii, and chords. Prove properties of angles for a quadrilateral inscribed in a circle. Objectives – To find the measure of an inscribed angle. To find the measure of an angle formed by a tangent and a chord.

Chapter 12.3 Notes Inscribed Angles – is an angle whose vertex is on a circle and whose sides contain chords of the circle. Inscribed angle is half the measure of the intercepted arc.

Thm – If 2 inscribed angles of a circle intercept the same arc, then the angles are congruent. Thm – If a rt is inscribed in a circle, then the hypotenuse is a diameter of the circle and vise versa.

Thm – A Quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary The Quad. is inscribed in the circle and the circle is circumscribed about the Quad.

Thm – If a tangent and an chord intersect at a pt Thm – If a tangent and an chord intersect at a pt. on a circle, then the measure of each angle formed is ½ the measure of its intercepted arc.

Chapter 12.4 Common Core – G.C.2 Identify and describe relationships among inscribed angles, radii, and chords. Objectives – To find measures of angles formed by chords, secants, and tangents. To find the lengths of segments associated with circles.

Chapter 12.4 Thm – If 2 chords intersect in the interior of a circle, then the measure of each angle is ½ the sum of the measures of the arcs intercepted by the angle and its vertical angles.

Thm – If a tangent and a secant, 2 tangents, or 2 secants intersect in the exterior of a circle, then the measure of the angle formed is ½ the difference of the measures of the intercepted arcs.

Thm – If 2 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. Thm – If 2 secant segments share the same endpt 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

Thm – If a secant segment and a tangent segment share an endpt Thm – If a secant segment and a tangent segment share an endpt. Outside a circle, then the product of the length of the secant segment and the length of its external segment equals the square of the length of the tangent segment.

Chapter 12.5 Notes Common Core – G.GPE.1 Derive the equation of a circle given center and radius using the Pythagorean Theorem. Objectives – To write the equation of a circle. To find the center and radius of a circle.

Chapter 12.5 Notes Standard equation of a circle (x – h)2 + (y – k)2 = r2 Center (h,k) Radius is r

Chapter 12.6 Notes Common Core – G.GMD.4 Identify three-dimensional objects generated by rotations of two-dimensional objects. Objectives – To draw and describe a locus.

Chapter 12.6 Notes Locus – is the set of all points in a plane that satisfy a given condition or a set of given conditions. Finding a Locus Draw any figures that are given in the statement of the problem. Locate several pts. that satisfy the given condition Continue drawing pts. Until you can recognize the pattern. Draw the locus and describe it in words.