Lesson 10-R

Chapter 10 Review

Objectives

• Review Chapter 10 material

Parts of Circles

• Circumference (Perimeter)– once around the outside of the circle; Formulas: C = 2πr = dπ

• Chord– segment with endpoints of the edge of the circle

• Radius– segment with one endpoint at the center and one at the edge

• Diameter– segment with endpoints on the edge and passes thru the center– longest chord in a circle– is twice the length of a radius

• Other parts– Center: is also the name of the circle – Secant: chord that extends beyond the edges of the circle– Tangent: a line (segment) that touches the circle at only one

point

Arcs in Circles• Arc is the edge of the circle between two points • An arc’s measure = measure of its central angle• All arcs (and central angles) have to sum to 360°• If two arcs have the same measure then the chords that form those

arcs have the same measure• If a radius is perpendicular to a chord then it bisects the chord and

the arc formed by the chord (example arc AED below)

• Major Arc (example: arc DAB)

– measures more than 180°

– more than ½ way around the circle

• Minor Arc (example: arc AED)

– measures less than 180°

– less than ½ way around the circle

• Semi-circle (example: arc EAB)

– measures 180°

– defined by a diameter

A

D

C 120°

BE is a diameterand AB = AD

B

E

120°

60°

60°

Angles Associated with Circles

NameVertex

LocationSides Formula Example

Central Center radii = measure of the arc BCD = 110°

Inscribed Edge chords = ½ measure of the arc BAD = 55°

Interior Inside chords = average of the vertical arcs EVH = 73°

Exterior OutsideSecants / Tangents

= ½ (Big Arc – Little Arc)= ½ (Far Arc – Near Arc)

NVM = 30°

V

KL

MN

A

D

B

C 110°

minor arc BD = 110°

E

G

F

C 110°H

36° V

minor arc FG = 110°minor arc EH = 36°

minor arc LK = 10°minor arc NM = 70°

C

70°

10°

Segments Inside/Outside of Circles• Segments that intersect inside or outside the circle have the length

of their parts defined by:

J

J

K

KL

M

M

N

J

K

T

M

LJ · JM = NJ · JK3 8 = 6 4 JL · JN = JK · JM

5 12 = 4 15JT · JT = JK · JM

6 6 = 3 12

Two ChordsInside a Circle

Two SecantsFrom Outside Point

Secant & Tangent From Outside Point

L

N

Inside the circle, it’s the parts of the chordsmultiplied together

Outside the circle, it’s the outside part multiplied by the whole length OW = OW

4

68

3

63

9

5 4

117

Tangents and Circles• Tangents and radii always form a right angle• We can use the converse of the Pythagorean theorem to check if a

segment is tangent• The distance from a point outside the circle along its two tangents

to the circle is always the same distance

Example 1Given:JT is tangent to circle CJC = 25 and JT = 20

Find the radius

ST

J

C

Example 2Given:same radius as example 1JC = 25 and JS = 16

Is JS tangent to circle C?

JC² = JT² + TC²25² = 20² + r²625 = 400 + r²225 = r²15 = r

JC² = JS² + SC²25² = 16² + 15²625 = 256 + 225625 ≠ 481JS is not tangent

Equation of Circles• A circle’s algebraic equation is defined by:

(x – h)² + (y – k)² = r²

where the point (h, k) is the location of the center of the circleand r is the radius of the circle

• Circles are all points that are equidistant (that is the distance of the radius) from a central point (the center)

Summary & Homework

• Summary:– A

• Homework: – study for the test