Objectives: To use the relationship between a radius and a tangent To use the relationship between two tangents from one point

 Tangent to a Circle  a line in the plane of the circle that intersects the circle in exactly one point.  Point of Tangency  the point where a circle and a tangent intersect. A B Tangent to a Circle Point of Tangency

 Theorem 12.1 ◦ If a line is tangent to a circle, then the line is perpendicular to the radius drawn to the point of tangency. A B O AB ⊥ OP P

 Theorem 12.2 ◦ If a line in the plane of a circle is perpendicular to a radius at its endpoint on the circle, then the line is tangent to the circle. O A B P AB ⊥ ΘO

 Theorem 12.3 ◦ The two segments tangent to a circle from a point outside the circle are congruent. A B C O AB = CB ~

 Homework # 32  Due Tuesday (April 30)  Page 665 – 666 ◦ # 1 – 20 all

 Objectives: To use congruent chords, arcs, and central angles To recognize properties of lines through the center of a circle

 Chord  a segment whose endpoints are on a circle P Q O PQ and PQ

 Theorem 12.4 ◦ Within a circle or in congruent circles: 1. Congruent central angles have congruent chords 2. Congruent chords have congruent arcs 3. Congruent arcs have congruent central angles

 Theorem 12.5 ◦ Within a circle or in congruent circles: 1. Chords equidistant from the center are congruent 2. Congruent chords are equidistant from the center

 Theorem 12.6 ◦ In a circle, a diameter that is perpendicular to a chord bisects the chord and its arcs.  Theorem 12.7 ◦ In a circle, a diameter that bisects a chord (that is not a diameter) is perpendicular to the chord.  Theorem 12.8 ◦ In a circle, the perpendicular bisector of a chord contains the center of the circle.

 Homework #33  Due Wed/Thurs (May 1/2)  Page 673 – 674  # 1 – 19 all

 Objectives: To find the measure of an inscribed angle To find the measure of an angle formed by a tangent and a chord

 Inscribed Angle  angle in a circle in which the vertex is on the circle and the sides of the angle are chords of the circle  Intercepted Arc  arc created by drawing an inscribed angle A BC Intercepted Arc Inscribed Angle

 Theorem 12.9 – Inscribed Angle Theorem ◦ The measure of an inscribed angle is half the measure of its intercepted arc. A B C

 Corollaries to the Inscribed Angle Theorem ◦ 1. Two inscribed angles that intercept the same arc are congruent. ◦ 2. An angle inscribed in a semicircle is a right angle ◦ 3. The opposite angles of a quadrilateral inscribed in a circle are supplementary.

 Theorem ◦ The measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc. B D C B D C

 Homework #34  Due Friday (May 03)  Page 681 – 682 ◦ # 1 – 22 all

 Objectives: To find the measures of angles formed by chords, secants, and tangents To find the lengths of segments associated with circles

 Secant  a line that intersects a circle at two points. A B O

 Theorem ◦ The measure of an angle formed by two lines that intersect inside a circle is half the sum of the measures of the intercepted arcs. 1 x° y°

 Theorem cont. ◦ The measure of an angle formed by two lines that intersect outside a circle is half the difference of the measures of the intercepted arcs. 1 x° y°

 Theorem ◦ For a given point and circle, the product of the lengths of the two segments from the point to the circle is constant along any line through the point and circle.

a b P z y z y c d x w t a · b = c · d (w + x)w = (y + z)y

 Homework #35  Due Friday (May 03)  Page 691 ◦ # 1 – 14 all