na

1. MODERN GEOMETRY
Section 6.1
THOMPSON

2. ANNOUNCEMENTS
Return Exams

3. RATIOS / PROPORTIONS
Similar
 When objects have “the same shape”, but are not
congruent.
 Think about “scale model”
EFGABD  ~

4. RATIO
“a is to b”
ba
b
a
:

5. PROPORTION
“a is to b as c is to d”
dcba
d
c
b
a
:: 

6. VERIFYING PROPORTIONS
3
2
8.4
2.3


7. CROSS MULTIPLICATION
THEOREM
bcadiff
d
c
b
a

Proof…

8. SOLVING PROPORTIONS
148
5 x


9. SOLVING PROPORTIONS
GE
GF

10. GEOMETRIC MEAN
If
𝑎
𝑏
=
𝑏
𝑐
, then b is the geometric mean
of a and c.
Or… b = 𝑎𝑐
Find the geometric mean of 3 and 27

11. EXTREME AND MEAN RATIO
A B C
AB
AC
BC
AB
thatSuch

...
Construct B on AC

12. EXTREME AND MEAN RATIO
a b
b
a
a
ba
thatSuch


...

13. EXTREME AND MEAN RATIO
CONSTRUCTION
a b
Euclid’s Elements
Book VI – Proposition 30

14. D
A B
CM P
1. Construct square ABCD
2. Bisect DC for pt M
3. Extend side DC
4. Draw circle M radius MB intersect DC at P
5. Extend AB
6. Draw perpendicular to DC at P to AB
E

15. D
A B
CM P
Measure DC, CP
E
DC
DP
CP
DC
Does ?
What is the ratio?

16. EXTREME AND MEAN RATIO
ALTERNATE CONSTRUCTION
a b

17. EXTREME AND MEAN RATIO
TRIANGLE CONSTRUCTION
a b

18. FIBONACCI NUMBERS
1, 1, 2, 3, 5, 8, 13, 21, 34, 55,
89, …
Look at the ratio of Fn+1 / Fn

19. PHI – THE GOLDEN RATIO
(RHYMES WITH PIE)
a b


b
a
a
ba
thatSuch ...
But what is Phi?

20. PHI – THE GOLDEN RATIO
(RHYMES WITH PIE)
a b


b
a
a
ba
thatSuch ...
2
51


21. D
A B
CM P
E
THE GOLDEN RECTANGLE
AEPD

22. THE GOLDEN SPIRAL

23. THE GOLDEN SPIRAL

24. HOMEWORK
Page 303
 3 – 11 odd
 25, 27, 51
Read 6.2