1. Monday, 1/20, was MLK day – NO SCHOOL!!!!
1/20 Week
Monday, 1/20, was MLK day – NO SCHOOL!!!!

2. Tuesday – 1/21 Pickup late task and homework
We will be studying Circle Vocabulary
Handout graphic organizer and books
Section 9.1
Make notes on both, please.

3. Tuesday – 1/21
MCC9-12.G.C.2 Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
Youtube has a good demonstration at:
– NOTE: The central angle is simply an angle whose vertex is on the circle. It does not have to include the diameter

4. Wednesday, 1/22: Central Angle
A central angle is an angle whose vertex is at the center of the circle.
The sides of a central angle are two radii

5. Arcs A central angle cuts the circumference of a circle into two arcs.
The arcs are normally one minor and one major arc, but they could be the same measure if the central angle happened to be a diameter
Names and symbols of arcs
The measure of an arc is defined to be the measure of the central angle
Can also use the books definition of arcs with secants

6. Wednesday, 1/22
MCC9-12.G.C.5 Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.

7. What is a Radian? We have measured angles in degrees.
Go over estimates of angles in degrees.
A radian is another way to measure an angle
An angle measures one radian when the arc length is one radius long.

8. What is a Radian?
Use a plate to draw a large circle. Make sure the center of the circle is clearly marked. Use a straightedge to draw a radius of the circle.
Take a piece of string and “measure” the radius of the circle. Cut the string to exactly the length of the radius.
 Beginning at the end of the radius, wrap the cut string around the edge of the circle. Mark where the string ends on the circle.
Congratulations, you just measured a radian

9. What is a Radian?
Move the string to this new point and wrap it to the circle again. Continue this process until you have gone completely around the circle.
Roughly, how many radius lengths did it take to complete the distance around the circle?

10. Radians
Can you determine exactly how many radians there are in a circle?
Circumference = 2r
You are dividing the circumference into parts “r” long, which is what
Can you determine a way to convert from degrees to radians or from radian to degrees?

11. Repeat the above making a concentric circle inside the original circle, but a smaller radius
You have shown through similarity, that the radian measure should be the same no matter what the radius is.

12. Circle A: r = 7,  = 100 Central Angle Radius Arc Length (s) 100 7 7 
Central Angle in Radians
Arc Length (s)
Short Cut?
s = r

13. Circle A: r = 7,  = 100 Central Angle Radius Arc Length (s) 100 7 7 
Central Angle in Radians
Arc Length (s)
Short Cut?
s = r

14. Circle A: r = 7,  = 100 Central Angle Radius Arc Length (s) 100 7 7 
Central Angle in Radians
Arc Length (s)
Short Cut?
s = r

15. Circle A: r = 7,  = 100 Central Angle Radius Arc Length (s) 100 7 7 
Central Angle in Radians
Arc Length (s)
Short Cut?
s = r

16. Circle A: r = 7,  = 100 Central Angle Radius Arc Length (s) 100 7 7 
Central Angle in Radians
Arc Length (s)
Short Cut?
s = r

17. Circle A: r = 7,  = 100 Central Angle Radius Arc Length (s) 100 7 7 
Central Angle in Radians
Arc Length (s)
Short Cut?
s = r

18. Circle B:  = 100, arc length =
Central Angle
Radius
Arc Length (s)
Central Angle in Radians
Arc Length (s)
Short Cut?
s = r

19. Circle B: r = 10,  = 100, arc length =
Central Angle
Radius
Arc Length (s)
100
10 
Central Angle in Radians
Arc Length (s)
Short Cut?
s = r

20. Circle C: r = 12, arc length =
Central Angle
Radius
Arc Length (s)
Central Angle in Radians
Arc Length (s)
Short Cut?
s = r

21. Circle C: r = 12,  = 100, arc length =
Central Angle
Radius
Arc Length (s)
100
12 
Central Angle in Radians
Arc Length (s)
Short Cut?
s = r

22. Circle D: r = 15,  = Central Angle Radius Arc Length (s)
Central Angle in Radians
Arc Length (s)
Short Cut?
s = r

23. Circle D: r = 15,  = , arc length =
Central Angle
Radius
Arc Length (s)
100
15 
Central Angle in Radians
Arc Length (s)
Short Cut?
s = r

24. Summary
When circles have the same central angles, the ratio of corresponding linear parts is ____________
The Radian measure of the central angle is the ____________________
Explain why all circles are similar:________________________________
Simplify
Ratio of Radii
Ratio of Intercepted Arc Length

25. Summary
When circles have the same central angles, the ratio of corresponding linear parts is congruent
The Radian measure of the central angle is the ____________________
Explain why all circles are similar: _______________________________
Simplify
Ratio of Radii
Ratio of Intercepted Arc Length

26. Summary
When circles have the same central angles, the ratio of corresponding linear parts is a constant
The Radian measure of the central angle is the __arc length divided by the radius
Explain why all circles are similar: ________________________________
Simplify
Ratio of Radii
Ratio of Intercepted Arc Length

27. Summary
When circles have the same central angles, the ratio of corresponding linear parts is a constant
The Radian measure of the central angle is the __arc length divided by the radius
Explain why all circles are similar: the ratio of all corresponding linear dimensions is a constant, i.e., they have the same scale factor
Simplify
Ratio of Radii
Ratio of Intercepted Arc Length

28. HW
Skills page 728, 730, 731

29. Radians
Khan Academy