Transcript October 15, 2014

HW problem 1:
Use Geometer’s Sketchpad to construct a rectangle whose
side lengths are in the ratio of 2:1 without using the
perpendicular, parallel, or midpoint options in the construct
menu, and without constructing any circles.
HW Problem 2:
Construct the letter
A using Geometer’s Sketchpad.
A
Your
must be perfectly vertical and symmetric. In other
words, it cannot look like this
or this
.
HW Problem 3:
Construct a line segment AB using Geometer’s Sketchpad.
Without changing line segment AB in any way, construct a
rhombus (not a square) so that AB is a diagonal of the
rhombus.
4. Using Geometer’s Sketchpad, construct a regular pentagon with two
diagonals, as shown below.
a. Conjecture a relationship between the lengths of AC and CP. Prove your
conjecture.
b. Compute and display the ratio of BP to AP. Compute and display the
ratio of AB to AD. Compute and display the ratio of AD to AP.
c. Conjecture a relationship between the three ratios found in part (b).
d. Is there any special significance to the value of this ratio? Explain.
A
Conjecture:
All three ratios are equal.
C
P
D
B
5. A regular hexagon and a regular pentagon are given in the diagram. What
is the measure of ABC? It is not necessary to use Geometer’s
Sketchpad for this question.
132
A
C
B
6. Shown is a drawing of a kite. The drawing is
rotated 180 around point P and the rotation image
is then reflected over line m. Which of the following
represents the image after the reflection?
A.

P
m
C.

m
P
B.
P

m

m
D.

P
m
P
7. Recall problem 3 from Homework # 6:
A
“In the diagram, AB  CD , mADC = 63, mDCA = 41, and
mACB = 104. Compute the measure of ABC.”
Part of the solution was to “Flip triangle ABC so that
points C and A are reversed.” Use Geometer’s Sketchpad
B
to construct the diagram (with the correct angle measures)
C
and then verify the solution by “flipping” triangle ABC so
that points C and A are reversed and verifying that ABC  DAC.
Do NOT email your construction to me. Be prepared to share your method of
construction with the class next time.
D
8. A certain regular polygon has 90 diagonals. How many sides does the
polygon have?
15 sides
9. A certain regular polygon has angles that measure 165 each. How many
sides does the polygon have?
24 sides
10. In the polygon shown, sides AB, BC and CD are sides of a regular octagon,
and sides DE, EF, FG, and GA are sides of a regular decagon. Compute
the measure of angle BAG.
B
99
C
D
A
E
G
F
Definition:
Two triangles are similar if three angles of one are congruent to three angles of the
other, and their corresponding sides are in proportion.
C
F
12
ED EF DF


AB BC AC
9
B
E
D
DEF  ABC
scale factor 3 : 4
A
The ratio of any pair of
corresponding sides is
called the Scale Factor.
Definition:
Two triangles are similar if three angles of one are congruent to three angles of the
other, and their corresponding sides are in proportion.
C
F
ED EF DF


AB BC AC
B
E
D
A
Similar triangle theorems:
 Two triangles are similar if two angles of one triangle are congruent to two
angles of the other.
AA~
 Two triangles are similar if two sides of one triangle are proportional to
two sides of the other, and the angles between these sides are congruent.
SAS~
 Two triangles are similar if all three sides of both triangles are proportional.
SSS~
Can the triangles be proven similar? If so, state the similarity theorem that
supports the conclusion.
2.
1.
20
12
x
14
9
Yes, AA~
15
Yes, SAS~
What is the scale factor?
x = 23
1
3
3
5
Because ADE  ABC,
A
6
AD AE DE


AB AC BC
D
10
B
8
y
E
7.5
x  13
1
3
20
Theorem: If a line is parallel to one side of a triangle
and intersects the other two sides, it divides the other
two sides proportionally.
AD AE

BD EC
C
Three or more parallel lines divide
all transversals proportionally.
M
JK MN

KL
NP
a
=
J
b
K
L
N
a
b
P
Dilations
Use Geometer’s Sketchpad to construct an acute triangle ABC near the
lower left of the screen.
Construct a triangle similar to, but not congruent to ABC.
P
B
C
A
Dilations
Use Geometer’s Sketchpad to construct an acute triangle ABC near the
lower left of the screen.
Construct a triangle similar to, but not congruent to ABC.
A dilation is a transformation that
1. preserves angle measure and
2. changes lengths proportionally.
P
When a figure is dilated using scale
factor k, the image is k times as far
from the center of the dilation as the
original figure.
B
C
A
Another way to construct the midpoint of a segment
Question # 5 on the
Final Group Problem Solving Project
1.
What will happen to the height of the point
where the wires cross if the poles are moved
further apart or closer together?
KSU

1.
What will happen to the height of the point
where the wires cross if the poles are moved
further apart or closer together?
2.
How does the height of the point relate to the
heights of the flagpoles? (This is what
problem V is all about.)
KSU

PRACTICE WITH DILATION:
Use Geometer’s Sketchpad to construct right triangle ABC in which
mABC = 30 and mACB = 60. Then construct point D so that
AD = 13 (AB). Make a conjecture about how CD and DB are related.
Are there any similar triangles in this diagram? If so, name them.
B
D
A
C