GCSE Similarity Dr J Frost jfrosttiffin kingston sch

GCSE Similarity Dr J Frost (jfrost@tiffin. kingston. sch. uk) www. drfrostmaths. com GCSE Revision Pack References: 131, 137, 171, 172 Last modified: 7 th March 2017

GCSE Specification Pack Ref Description 171 Solve problems involving finding lengths in similar shapes. 172 Understand the effect of enlargement for perimeter, area and volume of shapes and solids. Know the relationships between linear, area and volume scale factors of mathematically similar shapes and solids 131 Convert between units of area 137 Convert between volume measures, including cubic centimetres and cubic metres

Similarity vs Congruence ! Two shapes are congruent if: They are the same shape and size ? (flipping is allowed) ! Two shapes are similar if: They are the same shape b (flipping is again allowed) b a a a b ?

Similarity 5 8 12 There’s two ways we could solve this: Scale factor approach? Two fractions approach ? “Cross multiply”

Basic Examples 1 2 17 11 10 40 18 15 Try using ‘scale factor’ method: Try using ‘two fractions’ method: ? ?

Harder Problems 3 In the diagram BCD is similar to triangle ACE. Work out the length of BD. Bro Tip: For some students it may help to draw out the two similar triangles separately. ? ? ?

Harder Problems In the diagram BCD is similar to triangle ACE. Work out the length of BD. 4 ? ? ?

Test Your Understanding 1 [Nov 2008 4 H Q 22] 2 The diagram shows a square inside a triangle. DEF is a straight line. What is length EF? (Hint: you’ll need to use Pythag at some point) ? ? ?

Exercise 1 (on provided sheet) 1 2 3 ? 4 ? ? 5 ? ?

Exercise 1 (on provided sheet) 8 6 [June 2014 1 H Q 20] Steve has a photo and a rectangular piece of card. ? 7 ? 9 ?

Exercise 1 (on provided sheet) 10 ? N 1 N 2 [IMO] A square is inscribed in a 3 -4 -5 right-angled triangle as shown. What is the side-length of the square? ?

Exercise 1 N 3 ? N 4 (on provided sheet) ?

A 4/A 3/A 2 paper puzzle… A 4 A 5 “A” sizes of paper (A 4, A 3, etc. ) have the special property that what two sheets of one size paper are put together, the combined sheet is mathematically similar to each individual sheet. What therefore is the ratio of length to width? ?

Similar Triangle Proofs ? ? November 2015 1 H Q 22 ? Required Knowledge: Circle Theorems

Test Your Understanding ?

Scaling areas and volumes A Savvy-Triangle is enlarged by a scale factor of 3 to form a Yusutriangle. 2 cm 6 cm ? 3 cm 9 cm ? Area = 3 cm? 2 Length increased by a factor of 3 ? Area increased by a factor of 9 ? Area = 27 cm ? 2

Scaling areas and volumes For area, the scale factor is squared. For volume, the scale factor is cubed. Example: A shape X is enlarged by a scale factor of 5 to produce a shape Y. The area of shape X is 3 m 2. What is the area of shape Y? Shape X Length: Shape Y Bro Tip: This is my own way of working out questions like this. You really can’t go wrong with this method! Area: 3 m 2 ? 75 m ? 2 Example: Shape A is enlarged to form shape B. The surface area of shape A is 30 cm 2 and the surface area of B is 120 cm 2. If shape A has length 5 cm, what length does shape B have? Shape A Length: Area: 5 cm 30 cm 2 Shape B ? ? ? 10 cm 120 cm 2

Scaling areas and volumes For area, the scale factor is squared. For volume, the scale factor is cubed. Example 3: Shape A is enlarged to form shape B. The surface area of shape A is 30 cm 2 and the surface area of B is 270 cm 2. If the volume of shape A is 80 cm 3, what is the volume of shape B? Shape A Length: Shape B Area: 30 cm 2 Volume: 80 cm 3 ? ? ? 270 cm 2 2160 cm ? 3

Test Your Understanding These 3 D shapes are mathematically similar. If the surface area of solid A is 20 cm 2. What is the surface area of solid B? B A Volume = 10 cm 3 Volume = 640 cm 3 Solid A Length: Solid B Area: 20 cm 2 Volume: 10 cm 3 ? 320 cm 2 640 cm 3 Answer = 320 cm 2

Exercise 2 1 (on provided sheet) Copy the table and determine the missing values. 4 Shape A Shape B Length: Area: Volume: 2 3 cm 5 cm 2 10 cm 3 ? ? 6 cm 20 cm 2 80 cm 3 ? ? Determine the missing values. Shape A Shape B Length: Area: Volume: ? ? ? 5 m 8 m 2 12 m 3 3 ? ? 15 m 72 m 2 324 m 3 ? 5 ? ? ?

Exercise 2 (on provided sheet) 6 7 ? ?

Exercise 2 (on provided sheet) 8 9 ? ? ? N ?

Units of Area and Volume We can use the same principle to find how to convert between units of volume and area. A unit (metre) square: The same square with cm: 1 m 100 cm ? ? ? ?

Further Examples ? ?

Exercise 3 1 ? 2 ? 3 ? 4 ? 5 ? 6 ? 7 8 ? ? ? ? ?