1. 933r
Properties of Two Similar Figures
Look at the figure of two coins below. Are they the
same in shape? Are they the same in size?
Figure 1.1
11..11
• scale factor
• similar
Key terms:
What are you going to
learn?
to decide two similar
or dissimilar plane
figures by mentioning
the properties
to determine the
length of unknown
side of two similar
figures
Look at the following shapes. Are they the same or different?
a. b.
c. d.
There are two figures in each set. Figures (b) and (c) are different in shape.
While, both (a) and (d) are the same in shape, but different in size.
Two figures which are the same in shape but different in size are said to be
similar figures.
Mathematics for Junior High School Grade 9 / 1

2. What are the properties of two similar figures?
Are quadrilaterals ABCD and EFGH below similar?
8 cm
A 9 cm B
6 cm
D 7.5 cm
C
6 cm
8 cm
E 12 cm
F
H
10 cm
G
Figure 1.2
The corresponding angles of quadrilaterals ABCD and EFGH are the same in
size.
∠ A = ∠ E , ∠ B = ∠ F , ∠ C = ∠ G , ∠ D = ∠ H.
The corresponding sides are proportional.
4
3
====
HG
DC
FG
BC
EF
AB
EH
AD
or
3
4
====
DC
HG
BC
FG
AB
EF
AD
EH
.
Because the corresponding angles have the same measure and the
corresponding sides are proportional, quadrilaterals ABCD and EFGH are
similar.
Two figures are similar if the corresponding angles have the
same measure and the ratio of the lengths of corresponding
sides is equal.
Look at the following two rectangles. One rectangle is 12 cm in length and 8
cm in width. The other one is 6 cm in length and 4 cm in width. Are they the
same? 12 cm
8 cm
6 cm
4 cm
Figure 1.3
2 / Student’s Book – Similarity and Congruency

3. All the angles of the previous rectangles are right angles, so the corresponding
angles have the same measure.
The ratio of the length is 2
6
12
= , and
The ratio of the width is 2
4
8
= .
Since they satisfy the properties of similarity of two planes, the two figures are
similar.
Look at the following triangles. Examine each corresponding sides. Are they
proportional? Are the two triangles similar?
R
13
Q
5
P
12
L 3
M
5
K
4
The triangle ABC on the right is an isosceles right-
angled triangle. AC = 20 cm. Which triangles are
similar to ∆EBD?
Mathematics for Junior High School Grade 9 / 3

4. The Length of an Unknown Side of Two Similar Figures
Observe the two similar polygons ABCDE and RSTUV below.
T
B
A
R
S
U
V
CD
E
9
x
6
4
5y
Figure 1.4
a What is the scale factor of polygon ABCDE to polygon RSTUV?
b What are the values of x and y?
Solution:
a The scale factor is the ratio of the lengths of two corresponding sides. This
means that:
9 3
6 2
AE
RV
= =
b Use ratio of corresponding sides to find x and y.
6 4
9
VR RS
EA AB x
= ⇔ = ⇔ 6x = 36 ⇔ x = 6
and
6 5
9
VR UT
EA DC y
= ⇔ = ⇔ 6y = 45 ⇔ y = 7.5
4 / Student’s Book – Similarity and Congruency

5. The three rectangles below are similar. Find the values of x and y.
E
F G
H
7.5
x
y
10 A
B C
D
8
3
1. For the following statements, circle T if the statement is always true, S if it is
sometimes true and F if it is always false.
a. T S F Two rectangles are similar.
b. T S F Two squares are similar.
c. T S F A triangle and a quadrilateral are similar.
d. T S F Two parallelograms are similar.
e. T S F Two equilateral triangles are similar.
f. T S F Two rhombuses are similar.
g. T S F Two regular pentagons are similar.
h. T S F Two isosceles triangles are similar.
i. T S F Two kites are similar.
2. Quadrilaterals RSTV and LMNO are similar. The sides of RSTV are 6 cm, 10
cm, 12 cm, and 14 cm in length, respectively. The shortest side of LMNO is
9 cm.
a. Determine the scale factor of RSTV to LMNO.
b. Find the length of other side of LMNO.
c. Find the perimeter of LMNO.
d. Find the ratio of the perimeter of RSTV to that of LMNO
Mathematics for Junior High School Grade 9 / 5

6. 3. Trapeziums ABCD and AEFG are
similar. ∠ AGF = 108o, GF = 14,
AD = 12, DG = 4.5, EF = 8 and
AB = 26. Determine the scale factor
of ABCD to AEFG.
Find:
a. (i) AG (ii) DC
(iii) ∠ADC (iv) BC
b. the perimeter of ABCD
c. the perimeter of AEFG
d. the ratio of the perimeter of ABCD to that of EFGA.
4. The two polygons below are similar. Find the values of x and y.
5. Reasoning Can you give an example of two dissimilar quadrilaterals
whose corresponding sides are proportional?
6. Reasoning Can you give an example of two dissimilar quadrilaterals
whose corresponding angles are the same size?
7. Reasoning Check whether two triangles whose corresponding sides are
proportional are similar.
12
A 15 B P Q
x
RS
C
T
12
15
E
y
D
24
16
E
FG
BA
D C
6 / Student’s Book – Similarity and Congruency

7. 8. Open Question The picture and its
frame on the right are similar. The length
of the picture is 80 cm, the length of the
frame is 100 cm, and the width of the
picture is 60 cm. Find the width of the
frame.
9. If the three pictures below are similar, determine the values of x and y.
3
6
x
y
4
6
10. Quadrilaterals ABCD and RSTU below are similar. Find the values of x, y, and z.
D
z
C
2
4
3
x
U
T
6
5
y
BA R S
11. Quadrilaterals RSTU and WXYZ below are similar. Find the values of a, b,
c, and d.
U
4
10
T
S
R
˚
˚
b
85˚
108
95
6
W
a˚
X
6
d
˚
˚
85˚
108
95
c
3
Y
Z
Mathematics for Junior High School Grade 9 / 7

8. 12. Open Question A rectangular frame of photograph is 40 cm x 60 cm,
and a rectangular photograph is 30 cm x 40 cm. Are the frame and the
photograph similar? Suppose we modify the size of the frame so that the
frame and the photograph are similar. What is the size?
13. A rectangular folder is 25 cm x 35 cm and a piece of paper is 21 x 32 cm.
Are the folder and the paper similar? Modify the size of the paper so that
the paper and the folder are similar.
8 / Student’s Book – Similarity and Congruency