16.3 𝑘𝑚 (123.8 𝑐𝑚)
× 13000 (𝑡𝑜 2 𝑑. 𝑝)
How many times bigger is Ota than this in
real life?
Identifying Scales and
Ratios of Similarity
Slideshow 32, Mathematics
Mr. Richard Sasaki, Room 307
Objectives
• Recall some basic metric units for length
• Understand how to use a given scale using
ratio notation
• Recall necessary notation for similar shapes
• Understand how to find centres of
enlargement
Units
Let’s convert metric distances with units!
1 𝑚 = 100 𝑐𝑚
1 𝑘𝑚 = 1000 𝑚
220 𝑐𝑚
1400 𝑐𝑚
300000 𝑐𝑚
1700000 𝑐𝑚
50000 𝑐𝑚
7300 𝑐𝑚
850000 𝑐𝑚
3 𝑐𝑚
106000 𝑐𝑚
0.5 𝑚
18 𝑚
4000 𝑚
54 𝑚
70 𝑚
2500 𝑚
80 𝑚
110000 𝑚
2500 𝑚
4 𝑘𝑚
16 𝑘𝑚
0.5 𝑘𝑚
0.001 𝑘𝑚
0.08 𝑘𝑚
73 𝑘𝑚
8 𝑘𝑚
2 𝑘𝑚
15 𝑘𝑚
Scales
What is a scale?
A scale is a key (a plan) that we follow throughout
to make something smaller (or larger). Scales are
used to make maps and enlarge and shrink
appearances of objects.
Scales are normally in the form 1 ∶ 𝑎 when 𝑎 ∈ ℤ.
This image is the same
𝑎 is a number
size as my phone. 1 ∶ 1
that refers to how
much larger or
This image has the
smaller the object
dimensions halved. 1 ∶ 2
(or location)
Note: Ratios are not
actually is.
used for enlargement.
Models and Scales
Collectors models usually have a scale attached
to them. These are called scale models.
1 ∶ 32
1 ∶ 16
1∶8
As scales are usually lengths, not areas or volumes,
things appear to get much larger as 𝑎 decreases.
Map Reading
A map consistently follows the same scale so we
can calculate distances between locations as the
crow flies. (Without following roads, walkways etc.)
Note: We always measure from centre to centre.
This includes towns, other dwellings and structures.
The map has a scale of 1: 900.
Calculate the distance (in metres)
between Chonenji and Lawson.
49.5 × 900 = 44550 𝑐𝑚
44550 ÷ 100 = 445.5 𝑚
Note: Scales should have no
units. 1 𝑐𝑚 ∶ 1𝑘𝑚 = 1 ∶ 100,000
Answers
4 𝑐𝑚: 16 𝑘𝑚 ⇒ 4: 1,600,000 ⇒ 1: 400,000
𝑐𝑚 (𝑐𝑚)
2,000,000
1 ∶ (400,000 ÷ 1.4) = 1:
7
This value decreases as the map scale becomes
closer to real life.
2 𝑐𝑚 ⇒ 400,000 × 2 = 800,000 𝑐𝑚 ⇒ 8 𝑘𝑚
7 𝑐𝑚 ⇒ 400,000 × 7 = 2,800,000 𝑐𝑚 ⇒ 28 𝑘𝑚
2.5 𝑐𝑚 ⇒ 400,000 × 2.5 = 1,000,000 𝑐𝑚 ⇒ 10 𝑘𝑚
9.5 𝑐𝑚 ⇒ 400,000 × 9.5 = 3,800,000 𝑐𝑚 ⇒ 38 𝑘𝑚
Paper Size (Question 2)
𝑏
𝐴5
𝑎
2
___𝑏
× ?
𝐴4
(𝐴5 𝑎𝑟𝑒𝑎 × 2)
2
___𝑎
400,000 2
A5 Scale: 1 ∶ 400,000
=1∶
2
A4 Scale: 1 ∶ (400,000 ÷ 2) = 1 ∶ 200,000 2
Notation
Look at the statement below.
∆𝐴𝐵𝐶 ≅ ∆𝑋𝑌𝑍
This would be read as…
Triangle ABC is congruent to Triangle XYZ.
How would you read ∆𝐴𝐵𝐶 ~ ∆𝑋𝑌𝑍?
Triangle ABC is similar to Triangle XYZ.
So ≅ means congruent and ~ means similar.
Congruent (≅)
Similar (~)
Similar implies the
Congruent implies the
same proportions in
same size and shape.
size. The shape (angles)
Transposing, rotation and
must be the same.
reflection are accepted.
Answers
1𝑎. (A,) E, F
1𝑏. (A,) B, E, F, G
𝐵
2. 𝐴𝐵 = 3𝐴′𝐵′
𝐵𝐶 = 3𝐵′ 𝐶′
𝐶𝐷 = 3𝐶 ′ 𝐷′
𝐶
𝐷
𝐷𝐴 = 3𝐷 ′ 𝐴′
Well done if you remembered the line segment
symbols!
Don’t forget each of the following…
Line Segment AB is written as 𝐴𝐵.
Line AB is written as 𝐴𝐵 .
Ray AB (starting at A) is written as 𝐴𝐵 .
Similar Shapes
As you all know, similar shapes all have…
1. Equal Angles
1. Edges all in the same proportion
~
12𝑐𝑚
𝑜
65
65
10𝑐𝑚
𝑜
9.6𝑐𝑚 50𝑜
65𝑜 65𝑜
8𝑐𝑚
Like scales, similar shapes follow the same rules
throughout.
Centre of Enlargement
A centre of enlargement is a central point for
similarity. Two or more similar shapes can exist
where one is a transformation of another.
Example
Look at the image below.
Write down the transformed
version of edge 𝐸𝐴. 𝐸′𝐴′
If pentagon ABCDE is twice
the distance of it’s
transformation, write down
the transformation’s scale.
1: 2
Answers – Part 1
𝑂
𝐴’ 𝐵’
𝐷’ 𝐶’
𝑂𝐴 = 2.8 𝑂𝐴′ to 𝑂𝐴 = 3.2 𝑂𝐴′
1: 2.8 𝑡𝑜 1: 3.2
𝑥
2
𝑐𝑚
9
𝐴𝐷 = 2𝐵′ 𝐶′
𝐶′
𝐵′
𝑂
𝐴′
𝐴𝐵 = 2 3 𝐴′ 𝐵′ (𝐴𝐵 = 0.6 𝐴′ 𝐵′ to 𝐴𝐵 = 0.75 𝐴′ 𝐵′)
No centre of
enlargement
The transformation is double the base and height.
Area of ∆𝐴′ 𝐵′ 𝐶 ′ = 4𝑥 𝑐𝑚2