mathematical similarity
DESCRIPTION
Mathematical Similarity. Calculating Scale Factor. Similar Triangles. Parallel Line Triangles. Scale Factor in 2D (Area). Scale Factor 3D (Volume). www.mathsrevision.com. Exam Questions. Starter Questions. Q1.Solve4sin x – 1 = 0. Q2.Find the mini point for f(x) = (2 + x)(4 – x). - PowerPoint PPT PresentationTRANSCRIPT
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Level 4
MathematicalMathematicalSimilaritySimilarity
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Similar TrianglesParallel Line TrianglesScale Factor in 2D (Area)Scale Factor 3D (Volume)
Exam Questions
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Level 4
Friday 21 April 2023Friday 21 April 2023
Starter QuestionsStarter Questions
Q1. Solve 4sin x – 1 = 0
Q3. Factorise
Q4. Find the mean and standard deviation for the data
4d2 – 100k2
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Q2. Find the mini point for f(x) = (2 + x)(4 – x)
4 10 2 3 6
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Level 4
21 Apr 202321 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.
Learning IntentionLearning Intention Success CriteriaSuccess Criteria
1. We are learning what the term scale factor is and how it applies to shape.
1.1. Understand the term scale Understand the term scale factor.factor.
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2.2. Solve problems using scale Solve problems using scale factor.factor.
SimilaritySimilarity
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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 number 2 is true.
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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
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Given the shapes are similar, find the values y and z ?
Scale Factor applies to ANY SHAPES that are mathematically similar.
15 cm
3 cm
7 cm2cm
yz
6 cm
Scale factor = ESF = 3
15= 5
is 2 x 5 = 10
1.4 cm
Scale factor = RSF = 5
1
is 6 x 0.2 = 1.2
= 0.2
y z
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Given the shapes are similar, find the values a and b ?
Scale Factor applies to ANY SHAPES that are mathematically similar.
8 cm
7.5 cmb
10 cm
Scale factor = RSF = 10
4= 0.4
is 8 x 0.4 = 3.2
Scale factor = ESF = 4
10
is 2 x 2.5 = 5
= 2.5
a b
4 cm
2cm
a
3 cm
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Level 4
21 Apr 202321 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.
Now try TJ 4+
Ex 20.1Ch 20 (page 167)
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Scale FactorScale Factor
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Level 4
Friday 21 April 2023Friday 21 April 2023
Starter QuestionsStarter Questions
21 7 0x x
3 1
3 14 3
Q1. Find the roots to 1 decimal place
Q2. A freezer is reduced by 20% to £200 in a sale.What was the original price.
Q3. Calculate
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Level 4
21 Apr 202321 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.
Learning IntentionLearning Intention Success CriteriaSuccess Criteria
1. We are learning how the scale factor applies to similar triangles.
1.1. Understand how the scale Understand how the scale factor applies to similar factor applies to similar triangles.triangles.
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2.2. Solve problems using scale Solve problems using scale factor .factor .
Similar TrianglesSimilar Triangles
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Level 4
Similar Trianglesw
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70o 70o45o
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
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Level 4
Scale factors Scale factors
8cm12cm
Enlargement Scale factor?
Reduction Scale factor?
5cm
7.5cm
ESF =
RSF =
Can you see the relationship between the two scale factors?w
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1288
12
32
=
23
=
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Level 4
9cm a
Find a given ESF = 3
b
15cm
By finding the RSF Find the value of b.
ESF = 3
a9
=
Scale factors Scale factors w
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27cm
5cm
13
RSF =
b15
= = 515
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Level 4
21 Apr 202321 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.
Now try TJ 4+
Ex 20.2Ch 20 (page 169)
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Scale FactorScale Factor
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Level 4
Friday 21 April 2023Friday 21 April 2023
Starter QuestionsStarter Questions
1 11
4 2
Q1. A 42” TV is reduced by 10% to £540 in a sale.What was the original price.
Q3. Calculate
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Level 4
21 Apr 202321 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.
Learning IntentionLearning Intention Success CriteriaSuccess Criteria
1. We are learning how to use scale factor for parallel triangles.
1.1. Understand how the scale Understand how the scale factor applies to parallel factor applies to parallel triangles.triangles.
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2.2. Solve problems using scale Solve problems using scale factor for parallel triangles.factor for parallel triangles.
Parallel TrianglesParallel Triangles
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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
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A
B
C
DE
20 cm
45 cm
5 cm
y
The two triangles below are similar:
Find the distance y.
RSF = 50
5= AD
AE
10 1 x 20
y =
=
2 cm
10 1
=
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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 5
( ) 6 3
a SF
4.8 58 cm
3CD
5
( ) 3
b SF
4.5 57.5 cm
3AC
So perimeter
= 3 + 8 + 4 + 4.8
= 19.8 cm
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In a pair of similar triangles the ratio of the corresponding sides is
constant, always producing the same enlargement or reduction.
x
Corresponding sides are in proportion
ESFPQTS
=58
=
RT = x
=
x 6.4=
S
T
P
Q
R5
Find the values of x given that the triangles are similar.
8
6.4
58 x 4
4
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Level 4
21 Apr 202321 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.
Now try TJ 4+
Ex 20.3Ch 20 (page 171)
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Parallel TrianglesParallel Triangles
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Level 4
Friday 21 April 2023Friday 21 April 2023
Starter QuestionsStarter Questions
6, 5, 3, 1, 5
2 cos 1 0ox
Q1. Find the mean and standard deviation for the data
Q2. Solve the equation
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Level 4
21 Apr 202321 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.
Learning IntentionLearning Intention Success CriteriaSuccess Criteria
1. We are learning how the scale factor applies to area.
1.1. Understand how the scale Understand how the scale factor applies to area.factor applies to area.
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2.2. Solve area problems using Solve area problems using scale factor.scale factor.
Area of Similar ShapeArea of Similar Shape
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Level 4
It should be quite clear that second area is four times the first.
The scaling factor in 2D (AREA) is (SF)2.
x
y
x
y
Area = 2 x 2 = 4 Area = 4 x 4 = 16
For this example we have this case SF = 2 (2)2 = 4.ww
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Draw an area with sides 2 units long.
Draw an area with sides 4 units long.
2
24
4
Area of Similar ShapeArea of Similar Shape
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4cm
2cm
12cm
6cm
Small Area = 4 x 2 = 8cm2
Large Area = 12 x 6 = 72cm2
Connection ?
Another example of similar area ?
Work out the area of each shape and try to link AREA and SCALE FACTOR
Large Area = (3)2 x 8 =9 x 8 = 72cm2
Scale factor = ESF = 412 = 3
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ExampleThe following two shapes are said to be similar.If the smaller shape has an area of 42cm2.Calculate the area of the larger shape.
3cm
4cmWorking
ESF = 3
4
2
3
4
So area S.F =
Area of 2nd shape = 2
3
4
X 42 = 9
16X 42 = 74.67cm2
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Questions
1.
2.
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3.
4.
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Level 4
21 Apr 202321 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.
Now try TJ 4+
Ex 20.4Ch 20 (page 173)
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Area SimilarityArea Similarity
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Level 4
Friday 21 April 2023Friday 21 April 2023
Starter QuestionsStarter Questions
2, 4, 3, 1, 2
Q1. Draw a box plot to represent the data.
Q2. Find the coordinateswhere the line
and curve meet.
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Level 4
21 Apr 202321 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.
Learning IntentionLearning Intention Success CriteriaSuccess Criteria
1. We are learning how the scale factor applies to volume.
1.1. Understand how the scale Understand how the scale factor applies to 3D – factor applies to 3D – volume.volume.
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2.2. Solve volume problems Solve volume problems using scale factor.using scale factor.
Volumes of Similar SolidsVolumes of Similar Solids
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Level 4
4
44
Draw a cube with sides 2 units long.
Scale factor =
Using our knowledge from AREA section, (SF)2.
Volume = 2 x 2 x 2 = 8
For VOLUME the scale factor is (SF)3 = (2)3 = 8
Volume = 4 x 4 x 4 = 64x
y z
x
y zDraw a cube with sides 4 units long.
ESF = 2 4 = 2
2
22
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Volumes of Similar SolidsVolumes of Similar Solids
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Another example of similar volumes ?
Work out the volume of each shape and try to link volume and scale factor
3cm2cm
2cm
6cm4cm
4cm
V = 3 x 2 x 2 = 12cm3
V = 6 x 4 x 4 = 96cm3
Large Volume = (2)3 x 12 =
Scale factor = ESF = 3 6 = 2
Connection ?
8 x 12 =96cm3
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2cm
6cm
Given that the two boxes are similar, calculate the volume of the large box if the small box has a volume of 15ml
33 15
ESF =
So volume of large box = = 405 ml=
2 6
=
3
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23
2030
332 0.8
Example
ESF =
So volume of large jug = 278 0.8 = 2.7 litres=
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(a) SD : B = 4 : 3 SD : M = 8 : 5
34
(b) RSF =
23B 4Area 1600 900cm2
5
6
B
M(c) RSF
= 35
B 6Volume 2700 1562.5cm3
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3128 2
332 40
ESF =
So volume of large jug = 278 40 = £1.35=
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101 10
22.1544
(SF )3 =
So surface area ratio = (SF)2 = = 4.64
SF = 2.1544
Ratio of their surface area is 1 : 4.6 (to 1 d.p.)
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Level 4
21 Apr 202321 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.
Now try TJ 4+
Ex 20.5Ch 20 (page 175)
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Volume SimilarityVolume Similarity
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