Download Chapter 11: Extending Geometry

Chapter 11: Extending Geometry
11.3
Special Polygons
11.3.1. Golden Triangles and Rectangles
11.3.1.1. Golden Triangle
11.3.1.1.1. An isosceles triangle with the property that the ratio of the longer side to
the shorter side is the golden ratio
1+ 5
11.3.1.1.2. Golden Ratio: φ =
; φ ≈ 1.618
2
11.3.1.1.3. See figure in mini-investigation 11.6 page 691
11.3.1.1.4. α = 36o and β = 72o
11.3.1.1.5. 10 golden triangles present in the figure
11.3.1.2. Golden Rectangle
11.3.1.2.1. Considered an especially pleasing form
11.3.1.2.2. Seen in nature, buildings, works of art, and so on
11.3.1.2.3. To construct unit triangle
11.3.1.2.3.1.
Begin with unit square
11.3.1.2.3.2.
Find the midpoint of one side
11.3.1.2.3.3.
Construct an arc with a radius from the midpoint to a corner of the
square
11.3.1.2.3.4.
Extend the side of the square with the midpoint until it intersects the
arc
11.3.1.2.3.5.
Complete the rectangle using the newly formed longer side
11.3.1.2.3.6.
The distance from the midpoint of the square to the intersection of
5
the arc is defined as
2
11.3.2. Star Polygons
11.3.2.1. Using Circles to Produce Star Polygons
11.3.2.1.1. Star polygon
11.3.2.1.1.1.
interesting type of polygon
11.3.2.1.1.2.
frequently used in advertising logos, artistic designs, quilts, and
other decorative situations
11.3.2.1.2. Regular star polygon: vertices angles have equal measures and sides are
congruent
11.3.2.2. Generalization about Star Polygons
11.3.2.2.1. n = the number of equally spaced points on the circle
11.3.2.2.2. d = the dth point to which segments are drawn
n 
 n 
11.3.2.2.3. the star polygon   is the same as the star polygon 

d 
n − d 
n 
 n 
11.3.2.2.4. the figure produced for   and 
 is not a star polygon, but is a
1
n − 1
regular n-gon
n 
11.3.2.2.5. the n-sided star polygon   exists if and only if d ≠ 1, d ≠ n – 1, and n
d 
and d are relatively prime
a
n
n 
is equivalent to a lowest terms fraction , where   is a star
b
d
d 
a 
polygon, then   , sometimes called an “improper” representation of a star
b 
polygon, represents an n-sided star polygon
11.3.3. Star-Shaped Polygons
11.3.3.1. Star-shaped polygon
11.3.3.1.1. concave (nonconvex) symmetric figure that are NOT star polygons – see
figure 11.29 page 694
11.3.3.1.2. Star shaped polygons have: n star-tip points, 2n congruent sides, n
congruent point angles with measure α, and n congruent dent angles with
measure β
 360 
11.3.3.1.2.1.
Dent angles β = 
+α
 n 
 360 
11.3.3.1.2.2.
Point angles α = β − 

 n 
11.3.3.1.3. see figure 11.30 page 695 for comparison of star polygon versus starshaped polygon
11.3.4. Problems and Exercises p. 697
11.3.4.1. Home work: 2, 5, 7-10, 11, 12, 16, 17, 19, 20, 22, 23, 25, 27, 29
11.3.2.2.6.
if fraction