Dilations Investigation-Student Activity
Nrune:AnswerKey
Geometry
Dilations - Part I
1. On the picture below, draw rays from 0 through each vertex of MBC.
2. Use patty paper to measure the length of OA. Place A' on OA so that OA' = 2 • OA
3. Use the srune process to find B' and C'.
4. Draw M'B'C'.
5. Trace MBC on a piece of patty paper. Compare the angles of MBC with the angles of M'B'C' .
What do you notice about the angle measures?
6. Use a ruler to find the following lengths in centimeters.
a. AB= 3cm
d. A'B' =6 em
BC= 4cm
e. B'C'=8cm
b.
c. AC= Scm
f. A'C' =lOcm
7. What do you notice about the lengths of the sides of the two triangles?
The lengths of M 1 B 1 C 1 are twice as long as the lengths of MBC
8. The Scale Factor for this dilation is 2 or 2:1
0
Dilations Investigation-Student Activity
9. Use your patty paper to transfer MBC and M 1B 1C 1 to the grid paper with point 0 at the origin.
Write the ordered pairs next to each vertex on the grid paper. What do you notice about these ordered
pairs?
The coordinates of M1B 1C 1 are twice as big as the coordinates of MBC
10. Dilate MBC using a scale factor of3 on your grid paper. Label the new triangle MwBwCw. Write
the ordered pairs next to each vertex.
11. Dilate MBC using a scale factor of_!_ on your grid paper. Label the triangle M"'B"'C"'. Write the
2
ordered pairs next to each vertex.
Table 1- Ordered Pairs
Original
Triangle
Dilation
(Scale Factor of 2)
Dilation
(Scale Factor of3)
Dilation
I
(Scale Factor of-)
2
A
(3, 5)
A'
(6, 10)
A"
(9, 15)
A"'
(15, 25)
B
(3, 2)
B'
(6, 4)
B"
(9, 6)
B"'
(15, 1)
c
(7,2)
C'
(14, 4)
C"
(21, 6)
C"'
(3.5, 1)
12. What observations/conjectures can you make about dilations based on your work so far?
Possible answers:
When the scale factor is two,you multiply the coordinates by 2.When the scale factor is three, you
multiply the coordinates by three.When the scale factor is .!_,you multiply the coordinates by
2
13. Complete this conjecture based on Table 1: The image of
scale factor k has coordinates)m, kb ) .
(a, b)
.!_.
2
after a dilation with center 0 and
Dilations Investigation-Student Activity
Name:
Geometry
------------------------------------------------
Dilations - Part II
1. Record your observations about the characteristics of the triangles MBC and M'B'C' in the table
below.
M'B'C'
Observations
Original MBC
Characteristics
(Scale Factor of 2)
Coordinates
A(3, 5) B(3, 2) C(7, 2)
A(6,10) B(6,4)C(14,4)
Doubled
Angle Measures
53°-90°-37°
53°-90°-37°
Same measure
Length of Sides
3 em-4em-5 em
6 em--8 em-10 em
Doubled
Perimeter
12cm
24cm
Doubled
Area
6
24
Quadrupled
2. Graph Quadrilateral JKIM on the back of your grid paper. Dilate quadrilateral JKIM with a scale
£actor of 3 on your gn.d paper. Labelthe r.mage J IK' LIM I and comp11 e te the table be1ow.
Quad. l K L M'
Observations
Characteristics
Original JKIM
(Scale Factor of 3)
I (-6, 3) K1(-6, 9),
J(-2, 1) K(-2, 3),
Coordinates
Tripled
L1(9, 9), M'(9, 3)
L(3, 3), M(3, 1)
1
1
1
1
Angle Measures
90°- 90°- 90°- 90°
90°- 90°- 90°- 90°
Same measure
Length of Sides
2 emby5 em
6emby 15 em
Tripled
Perimeter
14cm
42cm
Tripled
Area
10cm2
90cm2
Multiplied by 9
3. Graph Quadrilateral QRST on the same grid paper. Dilate quadrilateral QRST with a scale factor of
! on your grid paper. Label the image
2
Q 1 R 1 S T 1 and complete the table below.
1
1
1
1
Q(-2,-2) R(-2, -4),
Quad. Q R S T
1
(Scale Factor of-)
2
Q1(-1, -1) R1(-1, -2),
S(-6, -4), T(-6, 2)
S1(-3, -2), T'(-3, 1)
Angle Measures
135°- 90°-90°-45°
135°- 90°- 90°- 45°
Same measure
Length of Sides
2 em-4 em-6 em-5.7 em
1 em-2 em-3 em-2.9 em
One half
Perimeter
17.7cm
8.8cm
One half
Area
16em2
4cm2
One fourth
Characteristics
Coordinates
Original QRST
1
Observations
One half
Dilations Investigation-Student Activity
4. A diameter of a circle has endpoints A and B. Graph the circle on your grid paper. Dilate the circle
W.it h a sea1e £actor of2 Labe l t he coord"mates of the I.mage A' and B' and compJ1ete the table below.
Original circle with
New circle with
Observations
Characteristics
- diameter AB
diameter A' B'
-
Coordinates
A(4,1), B(4,-3)
A'(8,2), B'(8,-6)
Doubled
Length of the diameter
4cm
8cm
Doubled
Length of the radius
2cm
4cm
Doubled
Circumference
41l" 12.56cm
Area
41l" 12.56cm
81l" 25.12cm
2
16.1l" 50.24 cm
Doubled
2
Quadrupled
5. Based on the information in all four tables, what conjectures can you make about:
a. Scale Factor used in the dilation and angle measures Scale factor has no effect on angle measure
Scale Factor used in the dilation and side lengths/diameter. Side length is multiplied by the scale factor
b.
c. Scale Factor used in the dilation and perimeter/circumference. Perimer is multiplied by the scale fact d.
Scale Factor used in the dilation and area.
Area is multiplied by the scale factor squared.
6. Use the graph to below to answer the following dilation questions.
[t-
'" " '""" -
-
'" " -
- i-
-
r-
,_
!-
f-
c
-- 1--- ---
-r
4 '""";;;iill'
" r"o
- j
-U-
·:k!-
f-
-- -
.... 0:,
·
.....
(r-
f-
'"""
-n.
-.t-1 - -f-- .
--- &
-1 -1 -- - 1 -- - 8
-i -1-
1-H-
a) Find the coordinates of triangle A' B'C' after a dilation
with a scale factor of 3.
A '(-9, 6) 8'(9,-6)C'(9,12)
How does this change angle measures?
-
Angles are the same
f-
1p
b) Find the coordinates of triangle A' B'C' after a dilation
with a scale factor of
1-
f--
1-
-
-10
'""" -!-
l_
2
A'(-4.5, 3)B'(4.5,-3)C'(4.5,6)
How does this change the lengths of the sides?
Side length is 3/2 bigger
c) If the coordinates of A' are (-1.5, 1) after a dilation, fmd the coordinates of B' and C'?
8'(1.5, -1)C'(1.5,2)
How does this change the perimeter of the triangle? ------'T:...!h e:..... p e""'nme·:::..:t::er"'---=is:.....:h al f a=s::....;b i g.
d) If the coordinates of B' are (18, -12) after a dilation, find the coordinates of A' and C '?
A'(-12,8) C'(12,16)
How does this change the area of the triangle?
2
The area is 6 or 36 times bigger
_
Dilations Investigation-Teacher resource
Name:
Geometry
-----------------------------------------------Dilations
v
/
,7
"''"'
"""
"'
I"'
J
7
7
)
v
"'""'
,7
"'
"'
/
b<
"'
[7
"""
v
v
/
v
v
v
v
v
A7
"' " 'k'
I
v
/
'
/
v
v
"'""""'
""""
/
[7
J
/
[7
"""
"
""""
/
-
,_,.,.....
'
_,...,.
""""
.
I
I
I
If
J
l/
,X
v
""" -
"""
"'........-
r
---
.,.....
u
. .....
.......
0
:::l
Vl
S'
<
(b
Vl
.....
..
.......
(IQ
.......
0
i
>-3
(b
::r
(b
v
v
v
v v
v
.........
/
""'
"""'
I
,/
I
I
J
\
'
........
...., V'i\
\
'
j
I
\
\
'
/
v
'"I
'"I
(b
Vl
0
(b
v