Section 9.5

9-5
1. Plan
Objectives
1 To locate dilation images
of figures
Examples
1 Finding a Scale Factor
2 Real-World Connection
3 Graphing Dilation Images
Math Background
Dilations map figures onto similar
figures. A dilation may or may not
have an invariant point that maps
onto itself but is uniquely
determined by its effect on any
two given points.
More Math Background: p. 468D
Lesson Planning and
Resources
See p. 468E for a list of the
resources that support this lesson.
PowerPoint
498
Bell Ringer Practice
Check Skills You’ll Need
For intervention, direct students to:
Ratios and Proportions
Lesson 7-1: Example 4
Extra Skills, Word Problems, Proof
Practice, Ch. 7
9-5
What You’ll Learn
• To locate dilation images of
figures
. . . And Why
To find the dimensions of a
car using the dimensions of
its model, as in Example 2
1 Locating Dilation Images
Real-World
Connection
Look closely at your pupil
in a mirror, and you can
watch it dilate.
1 A B C D E
2 A B C D E
3 A B C D E
4 A B C D E
5 A B C D E
B C D E
Test-Taking Tip
When finding a scale
factor, note first
whether the dilation
is an enlargement or
a reduction.
498 Chapter 9 Transformations
1
Dilations
Check Skills You’ll Need GO for Help
1
Lesson 7-1
2. 2 in. by 22 in.
1
Determine the scale drawing dimensions of a room using a scale of 4 in. ≠ 1 ft.
1. kitchen: 12 ft by 16 ft 3 in. by 4 in. 2. bedroom: 8 ft by 10 ft
3. laundry room: 6 ft by 9 ft
1 1
12 in. by 24 in.
New Vocabulary • dilation
4. bathroom: 5 ft by 7 ft
1 3
14 in. by 14 in.
• enlargement • reduction
A is a transformation whose preimage and image are similar. Thus, a
dilation is a similarity transformation. It is not, in general, an isometry.
Every dilation has a center and a scale factor n, n . 0. The scale factor describes
the size change from the original figure to the image.
To find a dilation with center C and scale factor n,
you can use the following two rules.
• The image of C is itself (that is, C9 = C).
• For any point R, R9 is on CR and CR9 = n ? CR.
The dilation is an enlargement if the scale factor is
greater than 1. The dilation is a reduction if the scale
factor is between 0 and 1.
B
C
F G
B
2
C 4
F
C
E
G
H
A= A D D
Enlargement
Center A, scale factor 2
E
Reduction
Center C, scale factor 1
4
H
)
dilation
n • CR
R
R
C = C
Center
EXAMPLE
Special Needs L1
Review the distinctions between similar and
congruent figures. Then show how a dilation needs
both a center and a scale factor by drawing lines that
pass through each vertex and the center of dilation.
Finding a Scale Factor
The blue triangle is a dilation image of the red
triangle. Describe the dilation.
The center is X. The image is larger than the
preimage, so the dilation is an enlargement.
XrTr
=
4 1 8
= 3
XT 4
The dilation has center X and scale factor 3.
R
R T
4
X X
Below Level L2
Before presenting dilations, review the definition of
similar polygons and the postulates and theorems
about similar triangles.
learning style: visual learning style: verbal
8
T

Z
Z
2
2
4
6
y
G
Quick Check
Quick Check
1 1
3. P9(1, 0), Z9(– 2 , 4),
1
G9( 2 , –1)
y
1
Z
x
1 O 2
1
G
P 4
x
P
G
P
Quick Check
1 The blue quadrilateral is a dilation image of the
red quadrilateral. Describe the dilation.
The dilation is a reduction with center
(0, 0) and scale factor .
1
2
2
3
Scale factors help you understand scale models, both large and small.
EXAMPLE Real-World Connection
Scale Models The packaging lists a model car’s length as 7.6 cm. It also gives the
scale as 1 ; 63. What is the length of the actual car?
To “enlarge” the model car to the actual car, use the scale factor 63. Multiply 7.6 cm
by 63 to get 478.8 cm, or about 4.8 m, for the length of the actual car.
2 The height of a tractor-trailer truck is 4.2 m. The scale factor for a model of the truck
1
is 54.
Find the height of the model to the nearest centimeter. 8 cm
Suppose a dilation is centered at the origin. You can find the dilation image of a
point by multiplying its coordinates by the scale factor.
Scale factor 4, (x, y) (4x, 4y)
4
2
O
y
A(2, 1)
A'(8, 4)
2 4 6 8
Advanced Learners L4
Students can use the definition of dilation, and
triangle similarity postulates and theorems, to prove
that the trapezoids above Example 1 are similar.
To dilate a triangle from the origin, find the dilation images of its vertices.
EXAMPLE
Graphing Dilation Images
Multiple Choice #PZG has vertices P(2, 0), Z-1,
1
2
, and
G(1, -2). What are the coordinates of the image of P for a
dilation with center (0, 0) and scale factor 3?
(5, 3) (6, 0)
(
2
3
, 0) (3, -6)
The scale factor is 3, so use the rule (x, y) S (3x,3y).
1
1 1
Scale factor 3,
(x, y) ( 3 x, 3 y)
y
O 2 4 6 8 x
2
B'(3, 2)
4
P(2, 0) S Pr(3 ? 2, 3 ? 0) , or Pr(6, 0) . The correct answer is B.
x
B(9, 6)
Z9 is 3 ? (21), 3 ? and G9 is (3 ? 1, 3 ? (22)) , so the vertices of the enlargement
at the left are P9(6, 0), Z9-3,
3
2
, and G9(3, -6).
3
1
Find the image of #PZG for a dilation with center (0, 0) and scale factor 2.
Draw the reduction. See left above.
1 2
learning style: verbal
6
y
2 O 2 4
M
3
M
J
J
K
L
Lesson 9-5 Dilations 499
2
y
Z P
2 1
2
English Language Learners ELL
Point out that a dilation scale factor is simply a
multiplier. The dilation is an enlargement for
a multiplier greater than 1, and a reduction
for a multiplier between 0 and 1.
G
K
x
x
L
learning style: verbal
2. Teach
Guided Instruction
Connection to Biology
Ask: When your pupil dilates, is
the dilation an enlargement or a
reduction? enlargement
3
EXAMPLE
Math Tip
Scalar multiplication is used only
for dilations centered at the
origin. Have students try to find
the image with center P to see
why this is so.
PowerPoint
Additional Examples
1 Circle A with a 3-cm diameter
and center C is a dilation of
concentric circle B with an 8-cm
diameter. Describe the dilation.
reduction with center C and
3
scale factor 8
2 The scale factor on a museum’s
floor plan is 1 : 200. The length
and width on the drawing are
8 in. and 6 in. Find the actual
dimensions in feet and inches.
133 ft, 4 in. by 100 ft
3 ABC has vertices A(-2, -3),
B(0, 4), and C(6, -12). What are
the coordinates of the image of
ABC for a dilation with center
(0, 0) and scale factor 0.75?
A(–1.5, –2.25), B(0, 3),
C(4.5, –9)
Resources
• Daily Notetaking Guide 9-5 L3
• Daily Notetaking Guide 9-5—
Adapted Instruction L1
Closure
Draw square ABCD, a dilation
image of ABCD with center A
and scale factor 3, and a dilation
image of ABCD with center C
1
and scale factor 2.
Check
students’ drawings.
499

3. Practice
Assignment Guide
1 A B 1-61
C Challenge 62-66
Test Prep 67-71
Mixed Review 72-81
Homework Quick Check
To check students’ understanding
of key skills and concepts, go over
Exercises 8, 16, 35, 42, 56.
Error Prevention!
Exercises 1–9 Students should be
careful not to describe dilations as
“as big as” or “bigger than” but
to give the precise scale factor.
Exercises 10–14 Ask students who
have model railroads to bring cars
to class to show other students.
Alternative Method
Exercises 15–17 Organize the
class into two groups. Have one
group copy the graphs and draw
segments to find the images
and the other group use scalar
multiplication to find the images.
Discuss the advantages of each
method.
GPS
500
Guided Problem Solving
Enrichment
Reteaching
Adapted Practice
Name Class Date
Practice 9-5
Find the area of each polygon. Round your answers to the nearest tenth.
1. an equilateral triangle with apothem 5.8 cm
2. a square with radius 17 ft
3. a regular hexagon with apothem 19 mm
4. a regular pentagon with radius 9 m
5. a regular octagon with radius 20 in.
6. a regular hexagon with apothem 11 cm
7. a regular decagon with apothem 10 in.
8. a square with radius 9 cm
Trigonometry and Area
Find the area of each triangle. Round your answers to the nearest tenth.
9. 10. 11.
12. 13. 14.
28 in. 32 in.
15. 10 cm
35
19 cm
16. 5 ft
65
4 ft
17. 15 m
46
59
63
6.5 m
13 m
9 mi
42
10 mi
38 18 km
10 km
34 in.
6 mm
46
54
26 in.
4.5 mm
Find the area of each regular polygon to the nearest tenth.
18. a triangular dog pen with apothem 4 m
19. a hexagonal swimming pool cover with radius 5 ft
© Pearson Education, Inc. All rights reserved. Practice
20. an octagonal floor of a gazebo with apothem 6 ft
21. a square deck with radius 2 m
22. a hexagonal patio with apothem 4 ft
15 m
L3
L4
L2
L1
L3
EXERCISES
For more exercises, see Extra Skill, Word Problem, and Proof Practice.
Practice and Problem Solving
A
Practice by Example
GO for
Help
Example 1
(page 498)
1. enlargement; center
A, scale factor
2. enlargement; center C,
scale factor 3
3. enlargement; center
R, scale factor
4. reduction; center K,
scale factor
5. reduction; center L,
scale factor
6. enlargement; center
M, scale factor 2
7. reduction; center (0, 0),
scale factor 1 3
2
3
2
1
3
1
3
2
Example 2
(page 499)
8. enlargement; center
(0, 0), scale factor 2
9. enlargement; center
(0, 0), scale factor 3 2
Example 3
(page 499)
500 Chapter 9 Transformations
The blue figure is a dilation image of the red figure. Describe the dilation.
1.
1–9. See left.
2. 3.
2
A
4. 5. 6.
7. y
8. y
9.
6
4
2
4
6
O 2 4
K
x
C
3
1
x
2
1
O 3
Model Railroads The table shows scales for
Model Railroad Scales
different types of model railroads. For each
model in Exercises 10–12, what would be the
Scale Name Scale Ratio
actual measurement?
121.94 in.
10. An HO-scale tank car is 1.4 in. high.
N
HO
1 : 160
1 : 87.1
11. An S-scale boxcar has length 8 in. 512 in.
S
1 : 64
12. A model of an engineer in a G-scale model O
1 : 48
train layout is 3 in. tall. 67.5 in.
13. A diesel engine is 60 feet long. How long is
its O-scale model? 1.25 ft
G
1 : 22.5
14. Actual railroad tracks are 4 ft 8.5 in. apart. How far apart are N-scale tracks?
0.35 in.
Find the image of kPQR for a dilation with center (0, 0) and the scale factor given.
Draw the image. 15–17. See back of book.
15. y Q
16. Q y 17.
Q y
2
x
P
4
2
O x
1
P
1
O 2 x
O
2
P
3
R
3 R
3
R
scale factor 3 scale factor 10 scale factor 3 4
L
9
R
4
4
2
M
y
4
O
x

B
Apply Your Skills
Exercise 41
27. Q9(–9, 12), W9(9, 15),
T9(9, 3), R9(–6, –3)
28. Q9(–6, 8), W9(6, 10),
T9(6, 2), R9(–4, –2)
A dilation has center (0, 0). Find the image of each point for the scale factor given.
18. D(1, -5); 2 D9(2, –10) 19. L(-3, 0); 5 L9(–15, 0) 20. A(-6, 2); 1.5
21. T(0, 6); 3 T9(0, 18) 22. M(0, 0); 10 M9(0, 0) 23. N(-4, -7); 0.1
24. F(-3, -2); 25. B
5
;
1
!3
4 10
26. Q6,
2
; !6
Find the image of QRTW for a dilation with center (0, 0)
y
W
and the scale factor given.
Q
4
27. 3 28. 2 29.
1
30.
1
2
4
2
31. 0.6 32. 0.9 33. 10 34. 100
T
3 O 2 x
35. Writing An equilateral triangle has 4-in. sides. Describe 1
R
its image for a dilation with scale factor 2.5. Explain.
36. Multiple Choice #D9E9F9 is a dilation of #DEF
with center (0, 0). Which of the following
statements is not true? B
#DEF and #D9E9F9 are similar.
m&F9D9E9 =
1
3
? m/FDE
The scale factor of the dilaton is
1
3
.
/EFD > /ErFrDr
, 22 A9(–9, 3)
1
3 F9(–1, – )
3
27–34. See margin.
The image has side lengths 10 in. and l measures 60.
y
D
4
E
6 4
D
E
2 O F
2
x
F
2
N9(–0.4, –0.7)
3
B9(
1
, –
1
8 15
)
3!2
Q9(6 !6,
2
)
Coordinate Geometry Graph MNPQ and its image 37–40. See back of book.
M9N9P9Q9 for a dilation with center (0, 0) and the scale factor given.
37. M(-1, -1), N(1, -2), P(1, 2), Q(-1, 3); scale factor 2
38. M(1, 3), N(-3, 3), P(-5, -3), Q(-1, -3); scale factor 3
39. M(0, 0), N(4, 0), P(6, -2), Q(-2, -2); scale factor
1
2
40. M(2, 6), N(-4, 10), P(-4, -8), Q(-2, -12); scale factor
1
4
41. Open-Ended Use the dilation command in geometry software or drawing
software to create a design that involves repeated dilations. The software
will prompt you to specify a center of dilation and a scale factor. Print
your design and color it. Feel free to use other transformations along
with dilations. Check students’ work.
42. Copy Reduction Your copy of your family crest is 4.5 in. wide. You need a
reduced copy for the front page of the family newsletter. The copy must fit in a
space 1.8 in. wide. What scale factor should you use on the copy machine?
Use a scale factor of .
A dilation maps kHIJ to kH9I9J9. Find the missing values.
43. HI = 8 in. 44. HI = 7 cm 45. HI = j ft 32
IJ = 5 in. IJ = 7 cm IJ = 30 ft
HJ = 6 in. HJ = j cm 12 HJ = 24 ft
H9I9= 16 in. H9I9= 5.25 cm H9I9= 8 ft
I9J9 = j in. 10 I9J9 = j cm 5.25 I9J9 = j ft 7.5
H9J9 = j in. 12 H9J9 = 9 cm H9J9 = 6 ft
2
5
46. Error Analysis Brendan says that when a rectangle with length 6 cm and width
4 cm is dilated by a scale factor of 2, the perimeter and area of the rectangle are
doubled. Explain what is incorrect about Brendan’s statement.
The perimeter is doubled but the area is multiplied by 4.
3 3 5
29. Q91– 2
, 22, W91 2 , 2 2,
3 1 1
T91 2 , 22,
R91–1, – 22
3 3 5
30. Q91– 4 , 12, W91 4 , 4 2,
3 1 1 1
T91 4 , 42,
R91– 2,
– 42
Lesson 9-5 Dilations 501
31. Q91–1.8, 2.42, W911.8, 32,
T911.8, 0.62, R91–1.2,
–0.62
32. Q91–2.7, 3.62, W912.7, 4.52,
T912.7, 0.92, R91–1.8, –0.92
Exercise 36 To help students
interpret the question correctly,
ask: What are you looking for?
The statement that is not true.
Teaching Tip
Exercise 41 Before students begin,
discuss how to create each figure
using dilations, rotations, and
translations.
Diversity
Exercise 42 Students unfamiliar
with heraldry may not know what
a family crest or coat of arms is.
Internet genealogy sites provide
many such examples for students
to examine.
Exercise 46 If necessary, review
Theorem 8-6: If the similarity
ratio of two similar figures is
a:b,then (1) the ratio of their
perimeters is a :b and (2) the
ratio of their areas is a 2 :b 2 .
Exercises 49–52 Exercises 49, 50,
and 52 can be done with compass
and straightedge. Exercise 51
requires a ruler.
Exercises 57–61 Use these
exercises for class discussion to
assess students’ understanding of
the lesson. If answers vary, have
students explain their reasons to
one another.
Connection to Film
Exercise 62 The ratio of the
perimeters of the similar triangles
is also 3 in. : 1 ft. Point out that
the flashlight models a film
projector, the small rectangle
models a film slide, and the large
rectangle models its projection
on a screen.
Visual Learners
Exercise 63 Suggest that students
place a tracing of AB and ArBr
over a coordinate grid.
33. Q91–30, 402, W9130, 502,
T9130, 102, R91–20, –102
34. Q91–300, 4002,
W91300, 5002,
T91300, 1002,
R91–200, –1002
501

4. Assess & Reteach
PowerPoint
502
Lesson Quiz
1. A model is a reduction
of a real tractor by the scale
factor of 1 : 16. Its dimensions
are 1.2 ft by
0.6 ft by 0.625 ft. Find the
actual dimensions of the
tractor. 19.2 ft by 9.6 ft
by 10 ft
For Exercises 2 and 3, kXYZ has
vertices X(3, 1), Y(2, –4), and
Z(–2, 0).
2. Use scalar multiplication to
find the image of XYZ for a
dilation with center (0, 0) and
scale factor 2.5. X(7.5, 2.5),
Y(5, –10), Z(–5, 0)
3. Draw and label the preimage
and image.
y
6
4 X'
2
Z' O X
Z
2 2 4 6 8
For Exercises 4 and 5, kDIL is a
dilation image of kDAT.
I
A
4
6
8
10
D
Y
2 m
T
Y'
6 m
4. Identify the center of dilation.
D
5. Find the scale factor. 4
Alternative Assessment
Have each student draw and label
a triangle with vertices A, B, and
C; choose a scale factor; and then
exchange papers with a partner
and draw the dilation image of
the partner’s triangle, using A as
the center. After drawing the
images, have partners compare
the pre-images and images and
check each other’s work.
L
GO
51.
B
T T
B A
Real-World
nline
Homework Help
Visit: PHSchool.com
Web Code: aue-0905
C
A
O
Connection
An overhead projection is a
dilation only when the mirror
in the head is tilted at a 458
angle. Turning the head
distorts the images.
Challenge
62. Each vertex is 1 ft
from the light.
63. Connect
corresponding points
A and A9 and B and
B9. Extend AA9 and
BB9 until they
intersect at the center
of dilation. The scale
factor is the length of
A9B9 divided by the
length of AB.
502 Chapter 9 Transformations
The diagram at the right shows kLMN and its
image kL9M9N9 for a dilation with center P.
47. Algebra Find the values of x and y.
48. Evaluate y and 2y - 60. Explain why the two
values must be equal. 60, 60; the two
Copy kTBA and point O for each of Exercises 49–52. Draw the dilation image
kT9B9A9 for the given center and scale factor.
49. center O, scale factor
50. center B, scale factor 3
49–50.
51. center T, scale factor See left.
52. center O, scale factor 2
53. Constructions Copy #GHI and point X onto your
paper. Use a compass and straightedge to construct
the image of #GHI for a dilation with center X and
scale factor 2. See margin.
Overhead Projection An overhead projector can dilate figures on transparencies.
54. A segment on a transparency is 2 in. long. Its image on the screen is 2 ft long.
What is the scale factor of the dilation? 12
55. The height of a parallelogram on the transparency is 4 cm. The scale factor is 15.
What is the height of the parallelogram on the screen? 60 cm
56. The area of a square on the screen is 3 ft2 x
1
2
T
1
3
B
O
A
I
G H
X
. The scale factor is 16. What is the
area of the square on the transparency?
Write true or false for Exercises 57–61. Explain your answers. 57–61. See
57. A dilation is an isometry. 58. A dilation changes orientation.
59. A dilation with a scale factor greater than 1 is a reduction.
60. For a dilation, corresponding angles of the image and preimage are congruent.
61. A dilation image cannot have any points in common with its preimage.
2 x 2
L
x ≠ 3; y ≠ 60
4 L
x
2
M
P y N
triangles are similar, so corresponding angles are congruent.
x + 3
M
N
(2y - 60)
See back
of book.
See margin.
ft
margin p. 503.
2 3
GPS
256
62. A flashlight projects an image of
B
rectangle ABCD on a wall so that each
B
C
vertex of ABCD is 3 ft away from the
A C
corresponding vertex of A9B9C9D9. The
A
D
length of AB is 3 in. The length of ArBr is 1 ft. See left.
D
not to scale
How far from each vertex of ABCD is the light?
63. Critical Thinking You are given AB and its dilation image ArBr with A, B, A9,
and B9 noncollinear. Explain how to find the center of dilation and scale factor.
See left.
Coordinate Geometry In the coordinate plane you can extend dilations to include
scale factors that are negative numbers.
a–c. See back of book.
64. a. Graph #PQR with vertices P(1, 2), Q(3, 4), and R(4, 1).
b. For a dilation centered at the origin with a scale factor of -3, multiply the
coordinates in part (a) by -3. List the results as P9, Q9, and R9.
c. Graph #P9Q9R9 on the same set of axes.
52. T
53.
B
B
T
A
A
O
G G
x
I
H
I
H

Problem Solving Hint
For Exercise 65(b),
recall the meaning of
reflection in a line.
Test Prep
Gridded Response
Mixed Review
GO for
Help
Lesson 9-5
Lesson 9-1
Lesson 8-5
lesson quiz, PHSchool.com, Web Code: aua-0905
57. False; a dilation doesn’t
map a segment to a O
segment unless the
scale factor is 1.
65. a. A dilation with center at the origin and scale factor -1 (see Exercise 64)
may be called a reflection in a point. For #PQR of Exercise 64, find the
image #P9Q9R9 for such a dilation. P9(–1, –2), Q9(–3, –4), R9(–4, –1)
b. Writing Explain why the dilation described in part (a) may be called
a reflection in a point. Extend your explanation to a new definition of
point symmetry. Compare your new definition with the definition given
on page 493. See margin.
66. Constructions Draw acute #ABC. Construct square DEFG so that DG is on
AC,
and E and F are on the other two sides of #ABC.(Hint: First, try the
special case with a right angle at A and use a dilation.) See margin.
67. A dilation maps #ABC onto #A9B9C9 with a scale factor of 0.3.
If A9B9 = 312 m, what is AB in meters? 1040
68. A dilation maps #CDE onto #C9D9E9. If CD = 7.5 ft, CE = 15 ft,
D9E9 = 3.25 ft, and C9D9= 2.5 ft, what is DE in feet? 9.75
69. A dilation maps #XYZ onto # X9Y9Z9. If XY = 4 m, YZ = 29 m,
X9Z9 = 28.7 m, and Y9Z9 = 29.145 m, what is X9Y9 in meters? 4.02
70. The center of dilation of quadrilateral ABCD
is point X, as shown at the right. The length
of a side of quadrilateral A9B9C9D9 is what
percent of the length of the corresponding
side of quadrilateral ABCD? 50
71. A dilation maps #JKL onto #J9K9L9.
If JK = 28 cm, KL = 52 cm, JL = 40.2 cm,
and J9K9 = 616 cm, what is the scale factor? 22
Coordinate Geometry A figure has a vertex at (–2, 7). If the figure has the given
type of symmetry, state the coordinates of another vertex of the figure.
72. line symmetry about the x-axis
(–2, –7)
74. point symmetry about the origin
73. line symmetry about the y-axis
(2, 7)
75. line symmetry about the line y = x
(2, –7)
Write a rule for each translation.
(7, –2)
76. 0 units to the right, 4 units up
(x, y) S (x, y ± 4)
78. 3 units to the left, 6 units down
77. 2 units to the left, 1 unit down
(x, y) S (x – 2, y – 1)
79. 8 units to the right, 10 units up
(x, y) S (x – 3, y – 6) (x, y) S (x ± 8, y ± 10)
For the angle of depression shown, find the value of x to the nearest tenth of a unit.
80.
25 ft
26
51.3 ft
81.
777.9 m
x
40
500 m
x
58. False; a dilation does
not change orientation.
59. False; a dilation with a
scale factor greater than
1 is an enlargement.
X
8
A
D
B
A
C
8
D
Lesson 9-5 Dilations 503
60. True; the image and
preimage are similar,
so the ' are O.
61. False; if the center
of dilation is on the
preimage, it is also
on the image.
B
C
Test Prep
A sheet of blank grids is available
in the Test-Taking Strategies with
Transparencies booklet. Give this
sheet to students for practice with
filling in the grids.
Resources
For additional practice with a
variety of test item formats:
• Standardized Test Prep, p. 527
• Test-Taking Strategies, p. 522
• Test-Taking Strategies with
Transparencies
65. b.Each point of the k is
reflected in the origin,
which is the point of
reflection. Two figures
are symmetrical with
respect to a pt. P if P
is the midpoint of
each segment that
connects two corr.
points of the figure.
66. Construct small square
D9E9F9G9 so that is
on (with D9 between
A and G9), E9 is on ,
and F9 is inside kABC.
Draw AF to meet BC at
F. Through F construct
the line n to AC.
Label
its point of intersection
with AB as E. Through E
and F construct the lines
# to AC.
Label their
points of intersection
with AC as D and G
respectively. DEFG is
the desired square.
S DrGr
AC
AB
9
503