The Geometry of a Circle - By: Dennis Kapatos

9. Teacher will tell students to construct an arc of a circle using the bottom of a can, or anything
else that will serve this purpose. The idea is that they don’t have the hole at the center like they
would from using a compass.
10. Teacher will ask students how they could find the center if they used a Mira. If students need
help, teacher will tell them to construct a point near both ends of the arc.
11. If students need more help, teacher will tell them to use their Mira find a position where it
maps one of these points onto the other.
12. If students still need more help, tell them to draw this line that the Mira is on top of when it
maps the two points onto each other.
13. Teacher will ask the students questions about what this line is and lead them to the idea that
this line must go through the center because it is a line of symmetry for the circle.
14. Students should recognize the need to repeat this to obtain another line and an intersection.
15. Teacher will ask students to connect the two pairs of points they mapped to each other with
segments at tell them that they are called chords.
16. Teacher will ask students to come up with a definition for a chord.
17. Teacher will ask what they constructed to the chord that found the center (a perpendicular
bisector to the chord, be definition of a perpendicular bisector).
18. Teacher will have students make conjecture of this (Theorem 12.8).
19. Teach will have student break groups, go to their computers, and start up GSP
20. Teacher will have student’s open a pre-made sketch of a circle, a chord, and its
perpendicular bisector.
21. Student’s will more around the points to see that any chord’s perpendicular
bisector crosses the center of the circle (see first picture).
22. Teacher will discuss with class and have students formalize their conjecture and
write Theorem 12.8 in their notes.
22. Teacher will have students look on anther page of this sketch of the satellite from
the beginning problem.
23. Teacher will ask students to find the placement of the receiver (see second
picture).
24. Teacher will students look at other pages of the sketches which will have them
discover Theorems 12.5, 12.6, and 12.9. They are very obvious and thus not too
much time is devoted to them. (see third and fourth pictures).
25. Students will enter these theorems into their notes.
26. Students will spend rest of class trying to solve question 20 on page 604.
?
Homework:
Length D'E on
m D'E = 4.83 cm
BC = 5.13 cm
E
drag
a a = 68.76 °
Prove Theorem 12.9, and on page 603 of the textbook, questions 1-13,
18, and finish 20.
G'
D'
B
Length FG' on BC =
m G'G = 4.83 cm
F
m AB = 3.27 cm
G
B
C
A
m CD = 3.3
m GE = 3.09 cm
E
F
D
m EF = 3.06 cm