SIMILARITY
Similar Polygons.
Note that the two polygons to the left differ in size but are
alike in shape.The two polygons are said to be similar.
A formal definition of similar (~) polygons includes terms such
as one-to-one correspondence, corresponding angles and corresponding sides.
Two polygons are said to be similar if and only if there is a one-to-one correspondence between their vertices such that:
1. Corresponding angles are congruent (equal (@))
2. Lengths of corresponding sides are in proportion.
The polygons shown are similar. When we say polygon CDEF is similar to polygon VWXY, we are asserting that the vertices have been paired as follows:
C V D W E X F Y Y X
1. Angle C @ angle V F E
Angle D @ angle W V
Angle E @ angle X W
Angle F @ angle Y C
D
2.
At this stage if we wish to show that two polygons are similar, we must establish that both conditions of the definition are met. The following figures show that meeting just one condition is not sufficient.
5 5
4 4
3 5
4 4
Polygon 1 Polygon 2 Polygon 3
Notice that although the corresponding angles of polygons 1 and 2 are congruent, the lengths of corresponding sides are not in proportion.
Now look at polygons 2 and 3. In this case the lengths of the corresponding sides are in proportion but the corresponding angles are not congruent. Again the polygons are not similar.
R 6 W A 12 D
3 4 6 8
S
2 T B
4
C
The perimeter of a polygon is the sum of the lengths of its sides. Consider the perimeters of the similar quadrilaterals above. Note that the ratio if the perimeter of the quadrilateral RSTW to the perimeter of quadrilateral ABCD is 15:30 or 1:2. How does this ratio compare with the ratio of the lengths of each pair of corresponding side?
1. If two polygons are similar, the ratio of their perimeter equals to the ratio of the lengths of any pair of corresponding sides.
Given: Polygon MORST is similar to polygon M’O’R’S’T’, the polygons have perimeter p and p’ respectively.
T’
T
M S M’ S’
O R
O’ R’
\
Exercise 1
Find the unknown sides for each of the following:
1. C
4 O
y x
5 B N
P 2
3 z
D M
6 C
2.
T
10
y S C 6
x 4 B
W 3
z R D A
7
3. I
8
y E
6
H D 5
z
3 F
J G
K x 6
4.
P x
O D 8
3 C
4
M 5 5
N 2
A
y
B
5. The drawing shows a rectangular picture 16cm × 8cm surrounded by a border of width 4cm. are the two rectangles similar?
6. The diagonals of a trapezium ABCD intersect at O. AB is parallel to DC, AB = 3cm and DC = 6cm. If CO – 4cm and OB = 3cm, find AO and DO.
7. From the rectangle ABCD a square is cut off to leave rectangle BCEF.
Rectangle BCEF is similar to ABCD. Find x and hence state the ratio of the sides of
rectangle ABCD. ABCD is called the Golden Rectangle and is an important shape in
architecture.
A F B
1
D 1 E C
Similar triangles.
At this point in our discussion of similar polygons, the only way that we can rove two triangles similar is by showing that the triangles satisfy the definition of similar polygons.
1. If two angles in one triangle are congruent to two angles of another triangle, the triangles are similar (AA)
Example:
B
D 1
E 4
3 2
C
F
Given: Plane figure with angle 1 @ angle 2
Angle 3 @ angle 4 because they are vertically opposite angles.
Because lengths of corresponding sides of similar triangles are in
proportion.
\D DEF ~ D CED
2. If an angle of one triangle is congruent to an angle of another triangle and the lengths of the sides including these angles are proportional, the triangles are similar.
C T
A B
R S
Given: D ABC and D RST with angle C @ angle T
\ D ABC ~ D RST
Example:
A tree of height 4m casts a shadow of length 6.5cm. Find the height of a house casting a shadow 26m long.
Tree House
6.5m 26m
4m x
26 × 4 = 6.5x
x = 16
Therefore, the height of the house is 16m.
Exercise 2
State whether the triangles are similar
1,
500 1000
300
2.
300
700
3.
300
a a b b
700 700
R
4. Complete the following question:
a) Triangle RST ~ Triangle ______
b) Complete the extended proportion
1
c) If RM = 3, MN = 4 and PS = 7, then ST = ______M N
d) If RM = 4, MN = 5 and ST = 8, then RS = ______
2
S T
5. Find the sides marked with letters in the following question.
A
Q
x 16cm y
6cm
B C P 3cm R
6cm
3. If a line is parallel to one side of a triangle and intersects the other two side, it divides them proportionally.
Given: Triangle ABC; C
Line YZ is parallel to line AB
Angle 1 @ angle 2; Y 2 4 Z
Angle 3 @ angle 4
1 3
A B
Example:
Given D ABC with line YZ parallel to line AB
Find the ratio of: C
a) CZ to ZB
7
Y Z
b) BC to ZC 4
A B
c) BC to BZ
Exercise 3
Find the sides marked with letters in questions 1 to 3; all lengths are given in centimeters.
1. 3. C
X w
3 4 B 6
B y D 3 4
2
A C A x E 2 D
10
2. P
m 5
Q 3 T
2 3
R S
Summary:
1. If two convex polygons are similar:
a) Corresponding angles are congruent
b) Lengths of corresponding sides are in proportion
c) The ratio of the perimeters equals the ratio of the lengths of any pair of corresponding sides.
2. Two triangles are similar if two angles of one triangle are congruent to tow angles of the other triangle.
3. In any triangle, a line that is parallel to one side and intersects the other two sides divides them proportionally.
4. It is important to note that some figures are a must to be similar. These figures include:
Ø Two equilateral triangles
Ø Two squares
Ø Two regular pentagons
Ø Two circles
Areas of similar shapes
The two rectangles are similar, the ratio of the corresponding sides
being k. A B W X
area of ABCD = ab a
area of WXYZ = ka × kb = k2ab ka
\ Area WXYZ = k2ab = k2 C b D
Area ABCD ab Z kb Y
This illustrates an important general rule for all similar shapes:
If two figures are similar and the ratio of corresponding sides k, then the ratio of their areas is k2
Note: k is sometimes called the linear scale factor
This result also applies for the surface area of similar three dimensional objects.
Area scale factor = (linear scale factor)2
Example 1:
XY is parallel to BC.
AB = 3 A
AX 2
If the area of triangle AXY = 4cm2, find the area of triangle ABC. 3 2
The triangles ABC and AXY are similar. X Y
Area scale factor = (linear scale factor) 2
4 = 22 B C
x 32
x = 9
Example 2:
Two similar triangles have areas of 18cm2 and 32cm2
respectively. If the base of the smaller triangle is 6cm,
find the base of the larger triangle.
Area scale factor = (linear scale factor)
18 = 62
32 x2
x = 8cm 6cm
x cm
Example 3:
A floor is covered by 600 tiles which are 10cm by 10cm. How many 20cm by 20cm tiles are needed to cover the same floor?
Total area = 10 × 10 × 600
= 60000cm2
\ For 20cm by 20cm, 60000
20×20
Therefore, 150 tiles are needed to cover the same floor.
Exercise 4:
1. Find the unknown area A. In each case the shapes are similar.
a)
4cm2 A
3cm 6cm
b)
2cm
3cm2 6cm A
c)
8cm 16cm
16cm
3cm
d)
A
27cm2
8cm
12cm
2. Find the lengths marked for each pair of similar shapes.
a)
5cm2 20cm2
4cm x
b)
3cm
z
c)
12cm2 5cm
3cm2 a
3. Given Ad = 3cm, AB = 5cm and area of triangle ADE = 6cm2 A
Find:
a) Area of triangle ABC
b) Area of DECB D E
B C
4. The triangles ABC and EBD are similar (AC and DE are not A
parallel)
If AB = 8cm, BE = 4cm and the area of triangle DBE = 6cm2, find
the Area of triangle ABC
D
B E C
5. A wall is covered by 160 tiles which are 15cm by 15cm. how many 10cm by 10cm tiles
are needed to cover the same wall?
6. When potatoes are peeled do you lose more peel or less when big potatoes are used as
opposed to small ones?
Congruency
Suppose we match the vertices of triangle ABC with those of triangle DEF in the following way
A D B E C F
This enables us to speak of correspondence between the triangles:
Triangle ABC Triangle DEF
C F
A B D C
In this correspondence, the first vertices named A and D, are corresponding vertices. So are the second and the third vertices named. Because A and D are corresponding vertices, angle A and angle D are called corresponding angles of the triangles. Other corresponding angles are angle B and angle E; angle C and angle F.
Because vertices A and B correspond to vertices D and E, line AB and line DE are called corresponding sides. Other corresponding sides are line BC and line EF; line AC and line
DF.
When the above six statements are true for triangle ABC and triangle DEF, the triangles
are said to be congruent ( @) triangles.
There are several ways to find if triangles are congruent. These include:
1. If three sides of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. (SSS)
C F
A
B D
E
According to postulate:
If line AB @ line DE, line BC @ line EF, and line AC @ line DF then
D ABC is @ D DEF.
2. If two sides and the included angle of one triangle are congruent to the corresponding parts on another triangle, the triangles are congruent. (SAS)
C
F
A B
D E
According to postulate:
If line AB @ line DE, line AC @ line DF and
angle A @ angle D then D ABC is congruent to D DEF
Exercise 5
1. Fill in the blanks:
a) Pair vertex A with vertex ______
b) Pair vertex C with vertex ______
c) Angle B and angle ______are corresponding angles.
d) Line CB and line _____ are corresponding sides.
e) Which statement is correct, D ABC @ D KSV or D ABC @ D KVS?
15 10
C B K S
7 10 7 15
A V
2. Prove two triangles are congruent.
a) Pair vertex D with vertex ______, and vertex C with vertex ______
b) Line DF and ______are corresponding sides.
c) Line EF and ______are corresponding sides.
d) Is the statement, D DEF @ D ZJO correct?
e) Is the statement, D OZJ @ D FDE correct?
F O
D 1100 1100 Z
E J
Some ways to find right triangles that are congruent are:
1. If two legs of one right triangle are congruent to the corresponding parts of another right triangle, the triangles are congruent. (LL)
T Z
R S X Y
D RST and D XYZ;
Angle S and angle Y are right angles.
Line RS @ line XY; line ST @ line YZ
\ D RST @ D XYZ.
2. If the hypotenuse and a leg of one right triangle are congruent to the corresponding parts of another right triangle, the triangles are congruent. (HL)
According to postulate:
If triangles RST and XYZ are right triangles with right angles S and Y, line RT @ XZ and line RS @ line XY then D RST @ D XYZ.
Exercise 6
1. In the figure it is given that line XC ^ line AE, line AX ^ line XD and line BX ^ line XE. Name a right triangle that has:
a) Hypotenuse XD X
b) Hypotenuse XE
c) Line XD as one of its legs
d) Line BX as one of its legs
e) Line AS as its hypotenuse.
f) Name every right triangle that has XC
as one of its legs.
A B C D E
2. Name each right triangle shown in the figure:
a) Given: Angle ACD is a right angle; line CD is perpendicular to line AB