Download Lecture 4 - UIUC Math

Math 117 Lecture 4 notes:
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.
The Seven Basic Euclidean Constructions
1. duplicate a line segment
2. duplicate an angle
3. bisect a line segment
4. perpendicular bisector of a line segment
5. perpendicular to a line at a point on the line
6. perpendicular to a line at a point not on the line
7. parallel to a line through a point not on the line
Other constructions you need to know:
1. angle bisector
2. dividing a line segment into equal parts
3. dividing a circle into given # equal parts
4. parallelogram, square, hexagon
note: most of lecture 4 is spent on constructions, and there may or may not be time to cover the
remaining material; however, the material is basic geometry and should be more or less a review rather
than new material for you.
Definition of Similarity:
Figures are similar if their corresponding angles are congruent and the lengths of their corresponding
sides have equal ratios. The ratios 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.
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.
Congruence and Similarity: 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 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
T
5
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.
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.
1. 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.
2. If a line divides two sides of a triangle into proportional segments, then the line is parallel to the
third side.
3. If parallel lines cut off congruent segments on one transversal, then they cut off congruent
segments on any transversal.
Midsegment Theorems (based on similarity and proportion):
1. The segment connecting the midpoints of two sides of a triangle is parallel to the third side and half
as long.
2. 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.
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. (AA)
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.
Triangle Congruence Theorems
Side, Side, Side Property (SSS)
Because a triangle is completely determined by its three sides, we have the following property:
If the three sides of one triangle are congruent, respectively, to the three sides of a second triangle,
then the triangles are congruent.
Side, Angle, Side Property (SAS)
If we know the lengths of two sides and the measure of the included angle, we can construct a unique
triangle. This property is:
If two sides and the included angle of one triangle are congruent to two sides and the included angle of
another triangle, respectively, then the two triangles are congruent.
Angle, Side, Angle Property (ASA)
If two angles and the included side of one triangle are congruent to two angles and the included side of
another triangle, respectively, then the triangles are congruent.
Angle, Angle, Side Property (AAS)
If two angles and a corresponding side of one triangle are congruent to two angles and a corresponding
side of another triangle, respectively, then the two triangles are congruent.
Other Theorems:
Triangle Inequality:
The sum of the measures of any two sides of a triangle must be greater than the measure of the third
side.
Any point equidistant from the endpoints of a segment is on the perpendicular bisector of the segment.
Any point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment.
The following hold for every isosceles triangle:
The angles opposite the congruent sides are congruent. (Base angles of an isosceles triangle are
congruent.)
The angle bisector of an angle formed by two congruent sides is an altitude of the triangle as well as the
perpendicular bisector of the third side.