Day 69: Circles

1. Day 69:Circles Round and round…

2. Definitions r P This is סּP • Circle • The locus (or set) of all points that are the same distance from a fixed point. This fixed point is called the center. A circle is named after its center. • Radius • The distance from the center to any point on the circle. Represented by r

3. Definitions C A B • Concentric • Two circles with the same center are called concentric. • סּA with radius AB isconcentric to סּA withradius AC • Congruent Circles • Two circles are congruent if and only if their radii are congruent.

4. Definitions • Diameter • A line segment with endpoints on the circle that passes through its center. Represented by d • Note that d = 2r, and r = ½d • Semicircle • Half of a circle. • A diameter divides a circle into two semicircles. • Circumference • The distance around a circle. In other words, its perimeter. Represented by C

5. Secants and Tangents and Chords (oh my!) • A secant is a line thatintersects a circle attwo points. • A tangent is a line thatintersects a circle atone point. • A chord is a line segmentwith both end points on a circle. • A diameter is a chord that passes through the center.

6. Definitions • Inscribed • A polygon is inscribed inside of a circle if all of its vertices lie on the circle. • Circumscribed • A circle is circumscribed about a polygon if it contains all of that polygon’s vertices. • A polygon is circumscribed about a circle if all of its sides are tangent to the circle. • Any triangle can have a circle circumscribed about it, or inscribed within it. • A quadrilateral can only be circumscribed and inscribed by circles if its opposite angles are supplementary (e.g., rectangles and isosceles trapezoids). • Convex kites can have circles inscribed in them, but cannot have a circle circumscribed about it. • Any other type of polygon must be regular to be inscribed and circumscribed by circles.

7. Pi (π) • All circles are similar, therefore all circles have the same proportions. Ancient mathematicians noticed that the ratio of a circle’s circumference and its diameter is the same for every circle. They designated this ratio as pi (π). • Thus • π = C/d • C = πd = 2πr • Pi is a non-terminal, non-repeating decimal. It is approximately 3.14159… • Unless otherwise stated, leave answers in “pi form” (not as a decimal).

8. Area • The area of a circle refers to the region bound by its circumference. • The area equals the square of the radius multiplied by pi: • A = π r2

9. B C A O D Angles and Arcs • An angle with its vertex at the center of a circle is called a central angle. Its sides contain two radii. • AOB, BOC, COD, etc. are all central angles. • The sum of the measures of the central angles of a circle with no common interior points is 360. • An arc is part of the circumference of a circle. • AB, AC, etc. are arcs.

10. B C A O D Angles and Arcs • We can refer to a minor or major arc. • A minor arc is less than 180. It is designated with two letters. • A major arc is greater than 180, and designated with three letters. It goes the ‘long way’ around the circle. • For example, there are two waysto get from A to C. • The short way is the minor arc AC. • The long way is the major arc ADC.

11. Semicircle A diameter divides a circle into two congruent arcs. Each of these is called a semicircle. Semicircles need three letters when we name them, because there are two of them.

12. B C A O D Angles and Arcs • There are two ways to measure arcs: by degrees and by length. • The arc measure is in degrees, and is the same as the central angle. • How many degrees are in a circle total? • How many degrees are in a semicircle? • The arc length is measured in feet, inches, meters, etc. The length will be a fraction of the total length of the circumference, based on the central angle divided by 360.

13. Theorem Two arcs, in the same circle or in congruent circles, are congruent if and only if their central angles are congruent.

14. B C A O D Arc Addition • Adjacent arcs are arcs in a circle that have exactly one point in common. • AB and BC are adjacent. • The measure of an arc formed by twoadjacent arcs is the sum of the measuresof the two arcs. • mAB + mBC = mAC

15. B C A O D Sectors • A sector of a circle is the area enclosed by a central angle and its arc. • Think ‘pie pieces’! • The area of a sector is a fraction of the circle’s area, based on the central angle divided by 360.

16. Theorem B A C P D • In the same circle (or congruent circles), two minor arcs are congruent if and only if their corresponding chords are congruent. • Given: AB  CDProve: AB  CD • When writing circle proofs, itwill be useful to remember thatall radii in a circle are congruent (bythe definition of radius).

17. Diameter/Chord Theorem • If a diameter or radius bisects a chord, it is perpendicular to the chord. • The converse is true: • If a diameter or radius is perpendicular to a chord, it bisects that chord. • OC bisects AB if and only ifOB  AB. • Note that if the diameter bisects the chord, it will also bisect the arc formed by that chord.

18. Theorem X B A C P Y D • In the same circle or congruent circles, two chords are congruent if and only if they are equidistant from the center. • Given: PX  PYProve: AB  CD

19. Homework 41-42 Workbook, pp. 124, 126, 127 Handout on sectors