16.3 𝑘𝑚 (123.8 𝑐𝑚) ×13000 (𝑡𝑜 2 𝑠.𝑓) How many times bigger is Ota than this in real life?

Identifying Scales and Ratios of Similarity Slideshow 32, Mathematics Mr. Richard Sasaki

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 7300 𝑐𝑚 50000 𝑐𝑚 1700000 𝑐𝑚 850000 𝑐𝑚 3 𝑐𝑚 106000 𝑐𝑚 0.5 𝑚 4000 𝑚 18 𝑚 54 𝑚 2500 𝑚 70 𝑚 80 𝑚 110000 𝑚 2500 𝑚 0.5 𝑘𝑚 4 𝑘𝑚 16 𝑘𝑚 0.08 𝑘𝑚 0.001 𝑘𝑚 73 𝑘𝑚 2 𝑘𝑚 8 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 when 𝑎∈ℤ. 1 :𝑎 𝑎 is a number that refers to how much larger or smaller the object (or location) actually is. This image is the same size as my phone. 1 :1 This image has the dimensions halved. 1 :2 Note: Ratios are not 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. cm 49.5×900=44550 𝑐𝑚 49.5 44550÷100= 445.5 𝑚 Note: Scales should have no units. 1 𝑐𝑚 :1𝑘𝑚=1 : 100,000

4 𝑐𝑚:16 𝑘𝑚⇒4:1,600,000⇒1:400,000 𝑐𝑚 (𝑐𝑚) 1 :(400,000÷1.4)=1: 2,000,000 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

𝐴3 𝐴4 Paper Size (Question 2) 2 ___𝑏 𝑏 (𝐴4 𝑎𝑟𝑒𝑎×2) × ? 𝑎 2 ___𝑎 × ? 𝑎 2 ___𝑎 =1 : 400,000 2 2 A4 Scale: 1 :400,000 A3 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 (~) Congruent implies the same size and shape. Transposing, rotation and reflection are accepted. Similar implies the same proportions in size. The shape (angles) must be the same.

Answers 1𝑎. (A,) E, F 1𝑏. (A,) B, E, F, G 𝐵 2. 𝐴𝐵 =3 𝐴′𝐵′ 𝐵𝐶 =3 𝐵 ′ 𝐶′ 2. 𝐴𝐵 =3 𝐴′𝐵′ 𝐵𝐶 =3 𝐵 ′ 𝐶′ 𝐶𝐷 =3 𝐶 ′ 𝐷′ 𝐶 𝐷 𝐷𝐴 =3 𝐷 ′ 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 ~ 9.6𝑐𝑚 50 𝑜 65 𝑜 65 𝑜 65 𝑜 65 𝑜 8𝑐𝑚 10𝑐𝑚 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

𝑂 𝑂𝐴=2.8 𝑂𝐴′ to 𝑂𝐴=3.2 𝑂𝐴′ 1:2.8 𝑡𝑜 1:3.2 𝑥 9 𝑐 𝑚 2 𝐴’ 𝐵’ 𝐷’ 𝐶’ 𝐴𝐷 =2 𝐵 ′ 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 𝑐 𝑚 2