Download Lecture 11 - UIUC Math

Math 103 Lecture #11 class notes
Math 103 Lecture 11 notes. Note: You will need scissors and ruler for this lecture. I will have a few flexible Safe-T
rulers for sale for $1 if you wish to buy one. When you get the handout in lecture, please cut out the circle labeled #1.
The Golden Ratio, ø = (1+√5)/2, also called the divine proportion, was known to the Pythagoreans in
500 B.C. and has many interesting applications in geometry. The golden ratio may be found using
the Fibonacci sequence; ratios of successive terms approach a decimal which is the golden ratio.
1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …
2/1
=
2
21/13 =
1.615
3/2
=
1.3
34/21 =
1.619
5/3
=
1.67
55/34 =
1.6176
8/5
=
1.6
89/55 =
1.618
13/8 =
1.625
144/89 =
1.618
A Golden Rectangle is one where the ratio of the length of the long side to that of the short side is
the Golden Ratio. The Ancient Greeks thought that the Golden Rectangle was the most aesthetically
pleasing of all. Along these line, notice how index cards are usually dimensioned 3x5 or 5x8, which
happens to be two pairs of numbers in the Fibonacci sequence whose quotients approximate ø.
What is the width of a rectangle with the property that if a square is cut out of the rectangle, the
remaining rectangle has the same length to width ratio as the original rectangle?
Fibonacci numbers and the Golden Section URL:
www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html
Geometric Constructions Historical note:
The straight line and circle were considered the basic geometric figures by the Greeks, and the
straightedge and compass are their physical analogs. It is believed that the Greek philosopher
Plato (427-347 B.C.) rejected the use of mechanical devices other than the straightedge and
compass for geometric constructions because use of other tools emphasized practicality rather
than "ideas" which he regarded as more important.
7 Basic Euclidean Constructions:
1.
duplicate a line segment
2.
duplicate an angle
3.
bisect a line segment
4.
construct the perpendicular bisector of a line segment
5.
construct a perpendicular to a line at a point on the line
6.
construct a perpendicular to a line from point not on the line
7.
construct a line parallel to a line through point not on the line
other constructions you should know:
8.
angle bisector
9.
incenter, circumcenter, orthocenter, centroid
10.
finding the center of a broken circle
In Math 117, we do these constructions using a compass & straightedge, and Geometer’s Sketchpad
software. In Math 103, we do these constructions using patty paper and mira’s.
Math 103 Lecture #11 class notes
Circles
Handout “Folding a Circle” to learn geometric terminology
Similar figure have the same shape.
Definition of Similarity:
Figures are similar if their corresponding angles are congruent and the lengths of their
corresponding sides have equal ratios (i.e., proportional). The ratio of corresponding lengths
is the scale factor of the figures.
Maps have scale factors.
Suppose a map shows a scale factor: 1" = 15 mi.
The distance between two cities measures 2 1/2 inches. How many miles apart are the
cities?
Finding Scale Factors:
A movie uses a model building that is similar to a real one. Find the scale factor from the
actual building to the model.
model
4 ft.
building
480 ft.
1 ft.
120 ft.
Definition of Similar Triangles: ΔABC is similar to ΔDEF (ΔABC ~ΔDEF) if and only if ∠A ≅ ∠D, ∠B ≅
∠E, ∠C ≅ ∠F, and AB/DE = AC/DF = BC/EF
1. In the following figure, find x:
P
12
8
x
R
5
T
S
M
2. For her art project, Rosa needed a wooden triangle that had one 18-inch side. The angles of the
triangle needed to have measures 36°, 84°, and 60°. Rosa had a carpenter make such a triangle. A
few months later, she needed another triangle with the same specifications. The first carpenter
was unavailable, so she contacted a different one. When the second triangle arrived, she was
surprised that it was not congruent to the first one. Explain how this is possible.
Angle-angle property of similar triangles: If two angles of one triangle are congruent, respectively,
to two angles of a second triangle, then the triangles are similar.
Math 103 Lecture #11 class notes
3. On a sunny day, a tall tree casts a 40-m shadow. At the same time, a meter stick held vertically
casts a 2.5-m shadow. How tall is the tree?
Theorems involving similarity and proportion.
• If a line parallel to one side of a triangle intersects the other sides, then it divides those
sides into proportional segments. The converse is also true.
• If a line divides two sides of a triangle into proportional segments, then the line is parallel
to the third side.
• If parallel lines cut off congruent segments on one transversal, then they cut off congruent
segments on any transversal.
Midsegment Theorems:
• The segment connecting the midpoints of two sides of a triangle is parallel to the third side
and half as long.
• If a line bisects one side of a triangle and is parallel to a second side, then it bisects the
third side and therefore is a midsegment.
Applying similarity to our Solar System . . .
if the SUN were a pumpkin about 1 foot in diameter,
Mercury would be a tomato seed about 50 feet away,
Venus would be a pea about 75 feet away,
Earth would be a pea about 100 feet away,
Mars would be a raisin about 175 feet away,
Jupiter would be an apple about 550 feet away,
Saturn would be a peach about 1025 feet away,
Uranus would be a plum about 2050 feet away,
Neptune would be a plum about 3225 feet away, and
Pluto would be smaller than a strawberry seed nearly a mile away.
Congruent
In math, the word similar (~) describes objects that have the same shape but not necessarily the
same size, while the word congruent ( " ) describes objects that have the same size as well as the
same shape. Whenever two figures are congruent, they are also similar. However, the converse is
not true. Definition of Congruence: figures are congruent if their corresponding angles and the
lengths of their corresponding sides are the same measures.
Definition of congruent segments and angles:
AB CD if, and only if, AB = CD
∠ABC " ∠DEF if, and only if, m(ABC) = m(DEF)
Definition of congruent triangles:
ΔABC " ΔDEF if, ∠A " ∠ D, ∠ B " ∠E, ∠C " ∠F, and AB = DE, BC = EF, AC = DF
The statement "corresponding parts of congruent triangles are congruent" is sometimes
abbreviated CPCTC.