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Arc measure in degees Arc length in units

Central angle

Inscribed angle

Chord

Radius

6.1 Circles Review Measuring an intercepted arc (measurement of the central angle created by that arc) D94 D94 ABC AB

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4

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TANGENT PROPERTIES

1. A tangent to a circle is perpendicular to the radius drawn to the point of tangency. 2. Tangent segments to a circle from a point outside the circle are congruent.

1. Tangent AT and tangent AN are tangents. m�ATN = 72° a = _________

a

T

N

A

2. The two line segments are tangents to the circle. Find x=________ x 140 3. Find the perimeter of the Quadrilateral.

A

B

C

O

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4

6.2 Properties of Chords. Theorem - in the same circle or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.

___________

CDAB # if and only if CDBA #�

Theorem - in the same circle or in congruent circles, two central angles are congruent if and only if their corresponding chords are congruent.

___________

CDAB # if and only if CODAOB �#� Theorem - if a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and it’s arc. Diameter = AB AB is perpendicular to CD, then AB bisects CD and arc CD Theorem - if chord AB is a perpendicular bisector of another chord, then AB is a diameter.

Theorem - In the same circle or in congruent circles, two chords are congruent if and only if they are equidistant from the center (perpendicular bisector determine the cords distance to the center).

AB is congruent to CD if and only if they are an equal distance away from P

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Follow the directions, answer the questions, and mark all congruent segments and angles. If two segments are perpendicular, then show they are

1. Draw the perpendicular bisector of _____

AB . 2. Mark where the perpendicular bisector intersects the circle (this segment is a diameter).

3. Draw the perpendicular bisector of _____

CD . 4. Mark where the perpendicular bisector intersects the circle (this segment is a diameter). Because any two congruent cords are equidistant from the center of the circle. 5. The intersection point of the perpendicular bisector is the center of the circle. Label it O. Does this seem familiar? _______center. 6. Draw radii OA, OB, OC, and OD. (are these segments congruent?) 7. Measure the central angles created by the cords AB and CD. (are they congruent?) 8. Two triangles are congruent, what conjecture says these two triangles are congruent? 1. Locate the center of the circle that contains aAB. 3. Draw a circle with a small cord and a larger cord. Which cord is closer to the center of the circle? What do you think you can say about the Diameter?

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6.3 Arcs and Angles Measuring an intercepted arc (measurement of the central angle created by that arc) D94 D94 Theorem - if an angle is inscribed in a circle, then it’s measure is half the measure of the of its intercepted arc(central angle). Theorem - if two inscribed angles of a circle intercept the same arc, then the angles are congruent.

w ithout the dotted lines Theorem - an angle that is inscribed in a circle is a right angle if and only if its corresponding arc is a semicircle . w ithout the dotted lines Theorem - a quadrilateral can be inscribed if and only if opposite angles are supplementary (the angles sum to 180°) Theorem- Parallel lines intercept congruent arcs on a circle.

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A

D R

G

1. m�P = _________ 2. b = _________

1 6 4 °

b

3. RE // PA c = _________ 4. d + e = _________

Give an example from the figure of each of the following. a. Tangent _________ b. Chord _________ c. Secant ________ d.Minor arc _________ e. Semicircle ________ 5. f = –?–. It's a tricky one, be careful.

aa

P

U

R

S 140°

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6.4 Proving Circle conjectures. Tangent Conjecture- A tangent to a circle is perpendicular to the radius drawn to the point of tangency. Tangent Segments Conjecture--Tangent segments to a circle from a point outside the circle are congruent. Chord Central Angles Conjecture-- If two chords in a circle are congruent, then they determine two central angles that are congruent. Chord Arcs conjecture-- If two chords in a circle are congruent, then their intercepted arcs are congruent. Perpendicular to a Chord Conjecture-- The perpendicular from the center of a circle to a chord is the bisector of the chord . Chord Distance to the Center Conjecture-- Two congruent cords in a circle are equidistant from the center of the circle. Perpendicular Bisector of a chord Conjecture--The perpendicular bisector of a chord passes through the center of the circle. Inscribed Angle conjecture-- The measure of an angle inscribed in a circle is one-half the measure of the intercepted arc. Inscribed Angles Intercepting Arcs conjecture-- Inscribed angles that intercept the same arc are congruent. Angles Inscribed in a Semicircle Conjecture-- Angles inscribed in a semicircle are right angles. Cyclic Quadrilateral conjecture--The opposite angles of a cyclic quadrilateral are supplementary. Parallel Lined Intercepted Arcs Conjecture-- Parallel lines intercept congruent arcs on a circle.

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6.5 Circumference and diameter ratios [Archimedes of Syracuse (287-212 BC)]

Circumference = rS2 or C = dS 𝜋 = 𝑡ℎ𝑒 𝑟𝑎𝑡𝑖𝑜 𝑜𝑓 𝑎 𝑐𝑖𝑟𝑐𝑙𝑒𝑠 𝑐𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑎𝑛𝑐𝑒 𝑡𝑜 𝑖𝑡𝑠 𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟.

2CrS

CdS

1. Find the circumference of a circle with radius 5 in. 2. Find the diameter of a circle with circumference S35 . 3. Find the diameter of a circle with radius 3 ft. 4. A circle is inscribed in a square of length 4 in. Find the circumference of the circle. (draw it) 5. Are two circles congruent if they have the same circumference?

Ptolemy (c. 150 AD) 3.1416

Zu Chongzhi (430-501 AD) 355/ 113 al-Khwarizmi (c. 800 ) 3.1416

al-Kashi (c. 1430) 14 places

Viète (1540-1603) 9 places

Roomen (1561-1615) 17 places

Van Ceulen (c. 1600) 35 places

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6.6 Around the world (applications of circumference) rC S2 1. Find the circumference.

r = 8.1 cm.. c | _________

r

Circumference is 6.53"

Length is 6.53" , or one revolution 2. What is the circumference of a wheel of radius 7cm? 4. How far will a wheel of radius 7 cm travel in 1 revolution? Distance | _________ 5. How far will a wheel of radius 7 cm travel in 11 revolutions? Distance | _________ 6. How many revolutions will a wheel of radius 3 ft make in 30ft? Revolutions | _________

7. If the diameter of the moon is 3475 km and an orbiting lunar station is circling 21 km above the lunar surface, find the distance traveled by the lunar station in one orbit. Distance | _________

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Circumference number of revolutionsTime�

Speed = Time

Distance= Example

HourMiles

8. A car traveled 3 miles in 1 hour. What was the average speed of the car? 9. A planet is orbiting the sun 42,000 miles from the center of the sun. It makes 1 revolution in 3 days. Find

the speed of the planet in miles per hour. 10. A Roach is orbiting a cheeto 4 inches from the center of the cheeto. It makes 1 revolution in 2 seconds.

Find the speed of the roach in inches per second.

11. Minute

sRevolutionrpm , example 3 rpm means 3 revolutions in 1 minute.

A wheel of diameter 4 ft is spinning at 4,000 rpm. How far does the wheel travel in 3 minutes? What is the speed of this wheel? A wheel of diameter 3 ft is spinning at 4,000 rpm. How far does the wheel travel in 3 minutes? What is the speed of this wheel? For the same revolutions, The bigger the wheel the _____________________the speed.

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D94

D94

7.7 Arc length Measuring an intercepted arc (central angle)

D360 D90 D240 A little review: Accurately shade the following:

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43

61

83

D180 D270 D60 D135

Arc Measure- Is a fraction of D360

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Arc Length ( the actual length of the Arc) - Is a fraction of the circumference of a circle.

nceCircumfereMeasureArc� D360

Here is why the formula is what it is:

ex. 360240

= of the circle.

D240

x The Arc Length is of the circumference.

x If the radius of the circle is 4 cm. Then the circumference is ______________. Arc Length = =

1. Arc length of aAB = _________ A

B 3 i n .

1 2 0 °

2. If the arc measure is 45 degrees and the radius is 4 inches, then find the arc length for the given arc measure. 3. If the arc measure is 120 degrees and the radius is 10 inches, then find the arc length for the given arc measure.

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4. r = 18 cm, arc length of aCD = _________

4 0 °

C D

r

5. If the arc length of GH = 8S cm. r = _________

G H

r

7 2 °

6. How far does the tip of a 6-inch minute hand on a classroom clock travel in 40 minutes? Draw a picture. Distance | _________

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Mixed review: