This document discusses similarity between shapes and triangles. It provides the following key points:
1. Two shapes are similar if corresponding angles are equal and corresponding sides are proportional. All regular polygons and rectangles can be similar or not similar depending on if they meet these criteria.
2. If two shapes are similar, one is an enlargement of the other. The scale factor represents the ratio of corresponding sides.
3. Similar triangles only require equal corresponding angles, not proportional sides. Unknown sides of similar triangles can be found using scale factors determined from corresponding sides.
4. Parallel lines drawn to the sides of a triangle form similar triangles. This property can be used to solve problems involving similar triangles.
Maths (CLASS 10) Chapter Triangles PPT
thales theorem
similar triangles
phyathagoras theorem ,etc
In this ppt all theorem are proved solution are gven
there are videos also
all topic cover
Maths (CLASS 10) Chapter Triangles PPT
thales theorem
similar triangles
phyathagoras theorem ,etc
In this ppt all theorem are proved solution are gven
there are videos also
all topic cover
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http://sandymillin.wordpress.com/iateflwebinar2024
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The Art Pastor's Guide to Sabbath | Steve Thomason
7th pre alg similarity & triangles
1. A
B
C
D
A B C D
Similarity
Similar shapes are always
in proportion to each
other. There is no
distortion between them.
2. Conditions for similarity
Two shapes are similar only when:
•Corresponding sides are in proportion and
•Corresponding angles are equal
All regular polygons
are similar
3.
4. Conditions for similarity
Two shapes are similar only when:
•Corresponding sides are in proportion and
•Corresponding angles are equal
All rectangles are not similar to one
another since only condition 2 is true.
5. If two objects are similar then one is an enlargement of the other
The rectangles below are similar:
Find the scale factor of enlargement that maps A to B
A
B
8 cm
16 cm5 cm
10 cm
Not to scale!
Scale factor = x2
Note that B to A
would be x ½
6. If two objects are similar then one is an enlargement of the other
The rectangles below are similar:
Find the scale factor of enlargement that maps A to B
A
B
8 cm
12 cm5 cm
7½ cm
Not to scale!
Scale factor = x1½
Note that B to A
would be x 2/3
7. 8 cm
If we are told that two objects are similar and we can find the
scale factor of enlargement then we can calculate the value of an
unknown side.
A
B
C
3 cm
Not to scale!
24 cm
x cm
y cm
13½ cm
The 3 rectangles are similar. Find the
unknown sides
Comparing corresponding sides in A and B: 24/8 = 3 so x = 3 x 3 = 9 cm
Comparing corresponding sides in A and C: 13½ /3 = 4½ so y = 4 ½ x 8 = 36 cm
8. 5.6 cm
If we are told that two objects are similar and we can find the
scale factor of enlargement then we can calculate the value of an
unknown side.
A
B
C
2.1 cm
Not to scale!
x cm
7.14 cm
26.88
cm
y cm
The 3 rectangles are similar. Find the
unknown sides
Comparing corresponding sides in A and B: 7.14/2.1 = 3.4 so x = 3.4 x 5.6 = 19.04 cm
Comparing corresponding sides in A and C: 26.88/5.6 = 4.8 so y = 4.8 x 2.1 = 10.08 cm
9. Similar Triangles
Similar triangles are important in mathematics and their
application can be used to solve a wide variety of problems.
The two conditions for similarity between shapes as
we have seen earlier are:
•Corresponding sides are in proportion and
•Corresponding angles are equal
Triangles are the exception to this rule.
only the second condition is needed
Two triangles are similar if their
•Corresponding angles are equal
10. 70o 70o
45o
65o
45o
These two triangles are similar since
they are equiangular.
50o
55o
75o
50o
These two triangles are similar since
they are equiangular.
If 2 triangles have 2 angles the
same then they must be equiangular = 180 – 125 = 55
11. Finding Unknown sides (1)
20 cm
18 cm 12 cm
6 cmb c
Since the triangles are equiangular they are similar.
So comparing corresponding sides.
18
6 12
b
=
6 18
9 cm
12
x
b = =
12
20 18
c
=
20 12
13.3 cm
18
x
c = =
12. 24 cm
10 cm 8 cm
6 cm
x
y
10
6 8
x
=
6 10
7.5 cm
8
x
x = =
8
24 10
y
=
24 8
19.2 cm
10
x
y = =
Comparing corresponding sides
13. Determining similarity
A B
E D
Triangles ABC and DEC are
similar. Why?C
Angle ACB = angle ECD (Vertically Opposite)
Angle ABC = angle DEC (Alt angles)
Angle BAC = angle EDC (Alt angles)
Since ABC is similar to DEC we know that corresponding
sides are in proportion
AB→DE BC→EC AC→DC
The order of the lettering is important in order to show which
pairs of sides correspond.
14. A
B C
D E
If BC is parallel to DE, explain why
triangles ABC and ADE are similar
Angle BAC = angle DAE (common to
both triangles)
Angle ABC = angle ADE (corresponding
angles between parallels)
Angle ACB = angle AED (corresponding
angles between parallels)
A
D E
A
D E
B
C
B
C
A line drawn parallel to any side of a triangle produces 2 similar triangles.
Triangles EBC and EAD are similar Triangles DBC and DAE are similar
15. A
B
C D
E
In the diagram below BE is parallel to CD and all
measurements are as shown.
(a) Calculate the length CD
(b) Calculate the perimeter of the Trapezium EBCD
4.8 m
6 m
4 m
4.5 m
10 m
A
C D8 cm 8 cm
3 cm
7.5 cm
10
( )
4.8 6
CD
a =
4.8 10
8 cm
6
x
CD = =
10
( )
4.5 6
AC
b =
4.5 10
7.5 cm
6
x
AC = =
So perimeter
= 3 + 8 + 4 + 4.8
= 19.8 cm
16. SF = 10/6 = 1.666..
AC = 1.666… x 4.5 = 7.5
∴Perimeter = 3 + 8 + 4 + 4.8 = 19.8 cm
10 m
A
C D8 cm 8 cm
3 cm
7.5 cm
So BC = 3 cm
So CD = 1.666… x 4.8 = 8 cm
A
B
C D
E
In the diagram below BE is parallel to CD and all
measurements are as shown.
(a) Calculate the length CD
(b) Calculate the perimeter of the Trapezium EBCD
4.8 m
6 m
4 m
4.5 m
Alternative
approach to the
same problem
17. A
B
C
DE
20 cm
45 cm5 cm
y
The two triangles below are similar: Find the distance y.
5
20 50
y
=
20 5
2 cm
5
x
y = =
18. Scale Factor = 50/5 = 10 So y = 20/10 = 2 cm
A
B
C
DE
20 cm
45 cm5 cm
y
The two triangles below are similar: Find the distance y.
Alternate
approach to
the same
problem
