Contents 1. Definition of Circle 2. PART of Circle 3. Properties of Circle 4. Circle Theorem
1. Definition of Circle A circle is a plane figure bounded by one curved line, and such that all straight lines drawn from a certain point within it to the bounding line, are equal. The bounding line is called its circumference and the point, its centre. —Euclid, Elements, Book I
2. PART of Circle 1. “Center” is the center of circle 2. “Radius” is the distance from the center to …..the circumference 3. “Diameter ” is the width of the circle that passesasse…..through the center 4. “Circumference” is the distance around the edge …..of a circle. 5. “Arc” is a fraction of the circumference. 1 - - - L
2. PART of Circle 6. “Chord” is a line joining two points on the circumference. 7. “ Secant” is an extended chord that cuts the circle at ……two distinct points. 8. “Tangent” is A line that touches the circumference of …..a circle at a point. 9. “Sector” is a region bounded by two radii of equal …..length with a common center. 10. “Segment” is the segment of a circle is the region …..bounded by a chord and the arc subtended by …..the chord. L
Semicircle Major Arc Minor Arc Central Angle Inscribed Angle Angle Inscribed in a semicircle 1. Relations of Central Angle, Arcs and Cords 3. Properties of Circle • • • L
In one circle, If two arcs are equal, then their corresponding central angles are equal, and their corresponding chords are also equal. arc chord central angle (Theorem 1) 1. Relations of Central Angle, Arcs and Cords 3. Properties of Circle 1 • 1 l 1 1 11 1 "
In one circle, If two arcs are equal, then their corresponding central angles are equal, and their corresponding chords are also equal. (Theorem 1) Example 1 arc length = 4 A B C D o If AC = CD and 1 = 45. Find the measure of 2 Textbook-Example 1 (Page 158) ..………………………………………………………………………… ..………………………………………………………………………… ..………………………………………………………………………… ..………………………………………………………………………… ..………………………………………………………………………… ..………………………………………………………………………… 3. Properties of Circle 1. Relations of Central Angle, Arcs and Cords 1ำ
In one circle, If two arcs are equal, then their corresponding central angles are equal, and their corresponding chords are also equal. (Theorem 1) Example 1 arc length = 4 A B C D o 2. If AB is the diameter, BC = CD = DE and BOC = 40. Find the measure of AOE Textbook-Practice (Page 159) ..………………………………………………………………………… ..………………………………………………………………………… ..………………………………………………………………………… ..………………………………………………………………………… ..………………………………………………………………………… ..………………………………………………………………………… E 3. Properties of Circle 1. Relations of Central Angle, Arcs and Cords ๏
3. Properties of Circle 2. Pythagorean Theorem in Calculating the Arcs Example 1 2. If OE l AB, the radius is 5, and OE = 3. Find the length of chord AB Textbook-Example 1 (Page 159) ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… A E B O A E O " l / , %. . " " " " "
3. Properties of Circle 2. Pythagorean Theorem in Calculating the Arcs Example 1 2. If the radius of circle O is 2 cm., the length of chord AB is 2 cm. Find the measure of AOB and the distance from O to AB Textbook-Example 2 (Page 160) ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… A C B O• i.㱺 i = % i ± s i %
3. Properties of Circle 2. Pythagorean Theorem in Calculating the Arcs Example 1 1. If the radius of circle O is 13., the length of chord AB is 24 cm. Find the distance from O to AB Textbook-Practice (Page 160) ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… A B O• ÷ \ s
3. Properties of Circle 2. Pythagorean Theorem in Calculating the Arcs Example 1 2. If AB is the diameter of circle O. Chord CD perpendicular bisects OB at E, CD = 4/3. Find the radius. Textbook-Practice (Page 160) ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… C B O D A \ • \ s
3. Properties of Circle 2. Pythagorean Theorem in Calculating the Arcs Example 1 3. Given the radius of circle O is 20 cm. AB is a chord in circle O, and AOB =. 120 Textbook-Practice (Page 160) ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… B O A ^ o • ± \ s
The angle formed by two line segment in (2) is call circumferential angle. 3. Properties of Circle 3. Circumference Angles (A. Properties of Circumferential Angle) (1) (2) (3) (4)
The circumferential angles corresponding to a semicircle or the diameter are all equal, which is a right angle, 90 . The arc that a 90 circumferential angle corresponds to is the diameter (Theorem 2) o 3. Properties of Circle 3. Circumference Angles (A. Properties of Circumferential Angle) A B C o A B C •- •-
o If line segment AB is the diameter of circle O. Point C is on circle. Then ACB is a circumferential angle formed by the diameter AB. What kind of angle could ACB be? Textbook-(Page 161) ..………………………………………………………………………… ..………………………………………………………………………… ..………………………………………………………………………… ..………………………………………………………………………… ..………………………………………………………………………… ..………………………………………………………………………… 3. Properties of Circle 3. Circumference Angles (A. Properties of Circumferential Angle) A B C ..………………………………………………………………………… ..………………………………………………………………………… ๏
In one circle, the measures of any circumferential angle of the same arc are equal and is one half of the measure of the central angle of that arc (Theorem 3) o 3. Properties of Circle 3. Circumference Angles (A. Properties of Circumferential Angle) B A C o A B C D = ญํ๊ Ee •- -
If AB is the diameter of circle O, and A = 80 . Find the measure of ABC Textbook- Example 1 (Page 162) ..………………………………………………………………………… ..………………………………………………………………………… ..………………………………………………………………………… ..………………………………………………………………………… ..………………………………………………………………………… ..………………………………………………………………………… 3. Properties of Circle 3. Circumference Angles (A. Properties of Circumferential Angle) o B A C 0 < ๓-
Given AB is the diameter of circle O, and D = 40 . Find the measure of CAB Textbook- Example 2 (Page 163) ..………………………………………………………………………… ..………………………………………………………………………… ..………………………………………………………………………… ..………………………………………………………………………… ..………………………………………………………………………… ..………………………………………………………………………… 3. Properties of Circle 3. Circumference Angles (A. Properties of Circumferential Angle) o A B C D / 0 ๏
Example 1 1. Given A, B, and C are points on circle O. ACB is a major arc. Which of the following has the same measure AOB A. 2C B. 4B C. 4A D. B + C Textbook-Practice (Page 163) ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… 3. Properties of Circle 3. Circumference Angles (A. Properties of Circumferential Angle) o B A C n n \ •
Example 1 2. The vertices of ABC, A, B, C are all on circle O. If ABC + AOC = 90, then AOC = Textbook-Practice (Page 163) ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… 3. Properties of Circle 3. Circumference Angles (A. Properties of Circumferential Angle) o B A C ^ o ^ \ •
Example 1 3. The Diameter of circle O, AB = 2, chord AC = 1. Point D is on circle O, then D = Textbook-Practice (Page 164) ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… 3. Properties of Circle 3. Circumference Angles (A. Properties of Circumferential Angle) o B A C D -
Example 1 3. The Diameter of circle O, AB = 2, chord AC = 1. Point D is on circle O, then D = Textbook-Practice (Page 164) ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… ..……………………………………………………………………………….…… 3. Properties of Circle 3. Circumference Angles (A. Properties of Circumferential Angle) o B A C D -