1. radius diameter secant tangent chord Circle: set of all points in a plane equidistant from a fixed point called the center. Circle 4.1

2. Theorem 1: In a plane, a line is tangent to a circle if and only if the line is perpendicular to a radius of the circle at the point of tangency. Theorem 2 : Tangent segments from a common external point are congruent. S R Q 32 3x + 5 32 = 3x + 5 27 = 3x 9 = x C B 50 r 70 r r 2 + 70 2 = (r + 50) 2 r 2 + 4900 = r 2 + 100r + 2500 2400 = 100r 24 = r Properties of Tangents 4.2

3. (1). In the same circle, or congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent. A C D B AB  CD if and only if (2). If one chord is a perpendicular bisector of another chord, then the first chord is a diameter. R S Q T Since SQ  TR and SQ bisects TR, SQ is a diameter of the circle. Properties of Chords4.3

4. (3). If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc. D H E GF If EG is a diameter and TR  DF, then HD  HF and GD  GF. Properties of Chords4.3 (4). In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center. E D A C BF G if and only if FE  GE

5. A B G If G is the center of the circle and m  AGB = 100 o, then mAB = 100 o. (1). The measure of a central angle is equal to the measure of its intercepted arc. (2). The measure of an inscribed angle is one half the measure of its intercepted arc. S R T If R is a point on the circle and m  TRS = 60 o, then mTS = 120 o. Central Angles & Inscribed Angles 4.4

6. (1). An angle inscribed in a semicircle is a right angle. (2). A quadrilateral can be inscribed in a circle, if and only if opposite angles are supplementary. B C A If BC is a diameter of the circle then m  CAB = 90 o. x°x° 88° 100° y°y° x o + 88 o = 180 o and y o + 100 o = 180 o x = 92 o y = 80 o Inscribed Angles 4.4

7. (1). The measure of the angle formed by two chords that intersect inside a circle is equal to half the sum of the measures of the intercepted arcs. 1 X Y Z W 80 o 60 o (2). The measure of an angle formed by 2 secants, 2 tangents, or a secant and a tangent is half the difference of the measures of the intercepted arcs. A C B D E 100 o 40 o 1 Angles of a Circle 4.5

8. (1). The rule for finding segment lengths formed by two chords is (part)(whole) = (part)(whole). x (2). The rule for finding segment lengths formed by two secants or a secant and a tangent is (outside)(whole) = (outside)(whole). Circles and Segments 4.6 6 10 3 x 5 7

9. (1). Circumference: C = 2  r or C =  d Circumference and Arc Length 4.7 (2). Arc Length: In a circle, the ratio of the length of a given arc to the circumference is equal to the ratio of the measure of the arc to 360 o. Example: 100 o 8 A B

10. (1). Area of a Circle: A =  r 2 Area of a Circle, Area of Sector 4.8 (2). The formula for the Area of a Sector is given by: Example: Let x represent the are of sector AB. 40 o 8 A C B

11. (1). Surface Area: A = 4  r 2 Surface Area and Volume of Sphere 4.9 (2). Volume: Example: Find the surface area and volume of a sphere whose diameter measures 14 cm.