10-1 circles i. definitions circle the set of all points in a plane that are at a given distance...

10-1 Circles I. Definitions Circle The set of all points in a

plane that are at a given distance from a given point in that plane. 

Symbol ○R Radius The distance between

the center of a circle and any point on the edge of the circle.  AR,BR

Diameter The distance across the circle that goes through the radius AB

Chord A segment that goes from one side of a circle to another.  AC

Circumference the distance around the circle C= 2 r

Total degrees 360

A

C

R

B

II. EXAMPLES

1. Find the circumference if

a. r = 8 b. d = 12

2. Find d and r if C = 136.9 cm.

3. Circle A radius=8, Circle B radius = 14, and JE =4. Find EB and DC.

DA J E B

C

10-2 Angles and Arcs10-2 Angles and Arcs

• I. Central Angle an angle whose

vertex contains the center of a

circle.

Part of a circle; the curve between two points on a circle.

II. ArcII. Arc

• If circle is divided into two unequal parts or arcs, the shorter arc (in red) is called the minor arc and the longer arc (in blue) is called the major arc.

A

BC

Minor arc- 2 letters

Major arc- 3 letters

•III. Semicircle– a semicircle is an arc that makes up

half of a circle

180°

Arc Addition Postulate - The measures of an arc formed by two adjacent arcs is the sum of the measures of the two arcs.   That is, if B is a point on      , then     +     =     . 

The measure of a minor arc = central angle. 

The measure of a major arc = 360 minus the measure of its central angle.

IV. ARC MEASURE

V. ARC LENGTHLENGTH OF THE ARC is a part of the circumference proportional to the measure of the central angle when compared to the entire circle

VII. CONCENTRIC CIRCLES

CONCENTRIC CIRCLES lie in the same plane and have the same center, but have different radii. 

ALL CONCENTRIC CIRCLES ARE SIMILAR BC ALL CIRCLES ARE SIMILAR!

VIII. CONGRUENT ARCS

TWO ARCS WITH THE SAME MEASURE AND LENGTH

Example 1: Find the length of Example 1: Find the length of arc RT and the measurement arc RT and the measurement

in degrees.in degrees.

2.2. a. Find the length of arcs RT a. Find the length of arcs RT and RSTand RST

b. Find the measurement in b. Find the measurement in degrees of both.degrees of both.

3. Find the arc length of RT 3. Find the arc length of RT and the degrees measurement and the degrees measurement

of RT.of RT.

4.4. If <NGE If <NGE < EGT, <AGJ =2x, <JGT < EGT, <AGJ =2x, <JGT = x + 12, and AT and JN are = x + 12, and AT and JN are diameters, find the following:diameters, find the following:

a. xa. x b. m NEb. m NE c. m JNEc. m JNE

A

J

T

N

E

5.

6. Find x.6. Find x.

RO

AQ

MN

9x

8x

19x

10-3 Arcs and Chords

I. Arc of the chord When a minor arc and a chord share the same

endpoints, we called the arc the ARC OF THE CHORD.

II. Relationships

– 2 minor arcs are if their chords are .

If a diameter is perpendicular to a chord, it bisects the chord and the arc.

2 chords are if they are equidistant from the center

Inscribed polygons must have vertices on the circle.

1. Circle N has a radius of 36.5 cm. Radius is perpendicular to chord FG, which is 53 cm long.

a. If m FG= 85, find m HG.

b. Find NZ.

2. Chords FG and LY are equidistant from the center. If the radius of M is 32,

find FG and BY.

FG = 46.4

LY = 23.2

3. mWX = 30, mXY = 50, mYZ = 30. WY= 14, FIND XZ.

4. RT is a diameter. If US = 9, find SV.

5. XZ= 12, UV = 8, WY is a diameter.Find the length of a radius.

6. IF AB and DC are both parallel and congruent and MP = 7, find PQ.

10-4 Inscribed Angles

• I. Definitions• Inscribed angle — An

angle that has its vertex on the circle and its sides are chords of the circle

• Intercepted arc — An intercepted arc is the arc that lies "inside" of an inscribed angle

• If an angle is inscribed in a circle, then the measure of the angle equals one-half the measure of its intercepted arc

If two inscribed angles of a circle or congruent circles intercept congruent arcs or the same arc, then the angles are congruent

If an inscribed angle of a circle intercepts a semicircle, then the

angle is a right angle

If a quadrilateral is inscribed in a circle, then its opposite angles are

supplementary

1.

2.

3. If mLM=120, mMN=45, and mNQ=105, find the numbered angles.

< 1= 22.5

< 2 = 60

< 3 = 45

< 4 = 22.5

< 5 = 112.5

4. If <2= 3a + 2 and < 3= 12 a – 2, find the measures of the numbered angles

m1 = 45, m4 = 45<2 = 20, < 3= 70

5. If mW = 74 and mZ = 112, find mY and mX.

68= mX

106 = mY

10-5 Tangents

• I. A line is TANGENT to a circle if it intersects the circle in EXACTLY ONE point.  This point is called the POINT OF TANGENCY.

If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.

II. Common External Tangents & Common Internal Tangents

A line or line segment that is tangent to two circles in the same plane is called a common tangent

COMMON

EXTERIOR

COMMON

INTERIOR

If two segments from the same exterior point are tangentto the circle, then they are congruent

ExamplesExamples• 1. AZ is tangent to O at point Z. Find x.

x

17.5

14

2. Determine whether BC is 2. Determine whether BC is tangent to tangent to AA. .

16.5 10

22

3. Determine whether is tangent 3. Determine whether is tangent to to HH. .

12

2418

4. Solve for x.4. Solve for x.

35 – 2 x

3 x - 5

5. Triangle 5. Triangle SCWSCW is circumscribed is circumscribed about about AA. Find the . Find the

perimeter of perimeter of SCWSCW if if WTWT = = 0.5(0.5(BCBC). ).

26

15

6. Find x so that the segment is a 6. Find x so that the segment is a tangent.tangent.

8x

10-6 Secants and 10-6 Secants and TangentsTangents

A SECANT is a line that intersects a circle in exactly two points. 

A SECANT is a line that intersects a circle in exactly two points. 

Every secant forms a chord

A secant that goes through the center of the circle forms a diameter.

If a secant and a tangent intersect at the point of tangency, then the measure of each angle formed is one-half the measure of its intercepted arc

If a secant and a tangent intersect at the point of tangency, then the measure of each angle formed is one-half the measure of its intercepted arc

If two secants intersect in the interior of a circle, then the measure of an angle formed is one-half the sum of the measures of the arcs intercepted by the angle and its vertical angle.

If two secants, a secant and a tangent, or two tangents intersect in the exterior of a circle, then the measure of the angle formed is one-half the positive difference of the measures of the intercepted arcs.

(     -    ) / 2 =  BVA (Big Arc - Little Arc) divided by 2

(      -    ) / 2 =  BVA     (Big Arc - Little Arc) divided by 2

5. 6.

7. 8.

9. 10.

11.

12.11.

13. 14.

10-7 Special Segments

I. ChordsIf two chords intersect in a circle, then the products of the measures of the segments of the chords are equal

(AO)(DO) = (BO)(CO).

If two secant segments are drawn to a circle from an exterior point, then the product of the measures of one secant segment and its external secant segment is equal to the product of the measures of the other secant segment and its external secant segment.

(EA)(EC) = (EB)(ED)

If a tangent segment and a secant segment are drawn to a circle from an exterior point, then the square of the measures of the tangent segment is equal to the product of the measures of secant segment and its external secant segment

(DC)2 = (DB)(DA)

1. 2.EXAMPLES: SOLVE.

3.

4

2x

X+ 3

4. 5.

6. 7.

8.9.

10. 11.

12. 13.

10-8 Equations of 10-8 Equations of CirclesCircles

10-8 Equations of 10-8 Equations of CirclesCircles

I. Graph of a circle• You can graph a

circle if you know:• Its center point

(h,k) • Its radius or

diameter

II. The equation• (x - h)2 + (y - k) 2 = r2

– Where (h,k) is your center point– And r is the radius

– Ex: (x - 3)2 + (y - 2) 2 = 42 or (x - 3)2 + (y - 2) 2 = 16

– So center is (3,2) and radius=4

Your turn• Ex 1: name the

center and the radius for:

• (x - 1)2 + (y -3) 2 = 25

Answer:• Center (1,3)• Radius 5 ( why? b/c 52 = 25 )

So, how do u write the equation?

• Center ( -1, 2) radius = 7

• remember : (x - h)2 + (y - k) 2 = r2

– So (x - -1)2 + (y - 2) 2 = 72 – (x +1)2 + (y - 2) 2 = 49

Your turn• Write the equation of a circle with

center (-5,-3) and diameter 16.

Answer• (x +5)2 + (y +3) 2

= 64

III. What if u are just given some points?

• Find an equation of the circle that has a diameter with endpoints at (6, 10) and (-2, 4).

• Step 1: Use the distance formula to find how long the diameter is:

• √(-2-6)2 + (4-10) 2

• =√100= 10 so radius is 5• Or just graph it an count how long it is!

• Step 2:• Graph it so you can see the center,

or find the half way point like this:– Take the x’s: 6+ -2 = 4 divide by 2=2– The y’s : 10+4= 14 then divide by 2 = 7– Center (2,7)

Finally!• Write the equation:(x -2)2 + (y -7) 2 = 25