Aim #5: How are dilations similar to and different from the other
transformations?
CC Geometry H
Do Now: 1) If ΔDEF is dilated by a scale factor of 4, which statement is true of
image ΔD'E'F'?
(1) 4D'E' = DE
(3) m≮D' = 4(m≮D)
(2) E'F' = 4EF
(4) 4(m≮F') = m≮F
2) CD is the image of AB after a dilation of scale factor k with center E. What
ratio would equal the scale factor of the dilation?
3) a. In a coordinate plane, where the center of dilation is the origin, to dilate a
point (x,y) by scale factor r, the dilated point will be located at (____, ____).
b. Let AB be a segment with endpoints A(-2,1) and B(1,-2). Dilate this segment
from the origin by a scale factor r = 4. What are the new coordinates?
A'_______B'________
c. Graph the segments.
d. Find the length of AB and A'B' in
simplest radical form. How do the
lengths compare?
e. Is segment length preserved under a dilation?
10°
273
4) If the scale factor that maps A
A' is 2, locate the center of dilation O, and
then P' using the same scale factor and center.
A'
A
P
5) a. Given center O and points A, B and C, find the points A', B' and C' using
center of dilation O and scale factor r = 3.*
b. Compare m≮ABC with m≮A'B'C': m ABC = _____ m A'B'C'= _____
Dilations preserve ____________________.
A
C
O
B
*A'= DO,3(A), B'= DO,3(B), and C'= DO,3(C).
The three rigid motion transformations are: _________________,
________________, and __________________.
Rigid motions preserve _________________ of segments and _______ measure.
How are dilations different from the three rigid motions?
• Dilations are NOT ______________ preserving, like the rigid transformations.
Dilations create distances that are scaled.
How are dilations similar to the rigid motions?
• Dilations are similar to rigid motions in that all transformations follow a rule
assignment, so any point P is assigned to a unique point F(P) in the plane.
• Dilations and rotations both require a center to define the function rule.
• There exists an inverse dilation that takes each point P' back to itself.
Example: A dilation of point P from center O by scale factor 2 has occurred. To
get back to the original point P, perform a dilation of ____ from center O.
6) Using a compass and straightedge, DO,2(PQ)
a) Is the dilated image a segment?
b) How has the pre-image been
transformed?
217
11°
7) Three cases to consider:
Case 1: A scale factor of a dilation is r = 1. Will a dilation from center O map PQ
to P'Q'? What happens in this case?
P
O
Q
Case 2: A scale factor of a dilation is r ≠ 1. Will a dilation from center O, which
lies on line PQ, map PQ to P'Q'? What happens in this case?
P
Q
O
Case 3: A scale factor of dilation is r ≠ 1. Will a dilation from center O,
which does NOT lie on line PQ, map PQ to P'Q'? ______
What are the possible values for r for the dilation
shown to the right? _____________
OP'= ___ x ____
OQ' = ___ x ___
OP' = OQ' =
Therefore:
OP OQ
Line segment P'Q' splits the sides of ΔOPQ _________________. By the
triangle side splitter theorem, we know that lines containing PQ and P'Q' are
____________.
8) Consider points P, Q, R, and S on a line, where P = R. Is there a dilation that
maps PQ to RS? Where is the center of dilation and why?
9) Given the dilation D0, 3/2 (PQ), and O which is not on PQ, what can you conclude
about the image of PQ?
P'Q' will be 1.5 times length of PQ and parallel to it.
10) Given figures A and B below, BA || DC, UV || XY, and UV ≅ XY. Determine
which figure has a dilation mapping the parallel line segments and locate the
center of dilation O. Determine which figure does NOT have a dilation and explain
why.
O
Dilation is occurring.
No dilation, UV and XY are || and congruent and therefore 2 sides of a para. Therefore, UX || XY and will never intersect at a center of dilation
Name __________________
Date ______________
CC Geometry H
HW #5
1) Which transformation is not considered a rigid motion? Why?
_____________________________________________________________
2) A dilation with center O and scale factor r takes A to A' and B to B'.
Locate center O and determine the scale factor.
3) Given center O and points A and B, find the points A'= DO,3 (A) and B'= DO,3 (B).
Measure and compare length A'B' with length AB by using a ratio. How does this
ratio compare to the scale factor?
OVER
Mixed Review:
X
Z
4) Given: VW ≅ UW, ≮X ≅ ≮Z
Prove: ΔXWV ≅ ΔZWU
Y
V
U
W
Statements
Reasons
H
5) A,B, and C are midpoints of the sides of ΔGHJ.
a) When AC = 3y - 5 and HJ = 4y + 2, what is HB? _____
A
B
G
C
J
2
b) When GH = x - 9 and CB = 4x, what is GA? _____
(1.5m+13)0
6) Find the value of each variable in the parallelogram.
4
x­
2u+2
5u­10
­x
14
(3m­8)0