S4 National 5 Notes Chapter 3

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1 S4 National 5 Notes Chapter 3 Similarity

2 SIMILARITY Representative Fractions A scale can be give as a ratio e.g. on a map 1: This is sometimes called the representative fraction. In the above example, 1 unit on the map stands for units on the ground. This unit can be in centimetres, inches, metres etc. As the representative fraction works for all units. Ex1 What length will represent a distance of 4km on a map with a scale of 1:50000? 4km = 4000m = 400,000cm Distance on map = 400,000 50,000 = 8cm Similarly: = km km = 0.08m 0.08 m = 8cm In order to work out a representative fraction we need to know one distance on the diagram and the actual distance. Ex The map distance from Beinn Tarsuinn and Beinn Bhreac is cm. The actual distance is 5 km. Find the representative fraction. cm :5km cm :500000cm 1 cm :50000cm 1:50000 Page of 1

3 Similar Triangles If two shapes are exactly the same shape and size they are said to be congruent. If two shapes are exactly the same shape but different sizes they are said to be similar. If two triangles are similar and we know which angles correspond we can work out the missing sides. Ex4 These two triangles are similar. Find the length of x. Solution Notice that the side marked x is an enlargement of the side marked 8. And that the side marked 9 is an enlargement of the side marked 3. Enlargement Scale Factor = Measurement From Larger Measurement From Smaller Page 3 of 1

4 Ex5 These two triangles are similar. Find x and y. Draw the triangles so that the corresponding triangles match up. x is an enlargement of the side marked 3 so: 0 Es.f = = 4 5 So x = Scale Factor 3 = 4 3 x = 1 y is a reduction of the side marked 14 so: Measurement From Smaller Reduction Scale Factor = Measurement From Larger 5 1 Rs.f = = = So y y = Scale Factor 14 = = 3. 5 Page 4 of 1

5 Similar Triangles Harder Examples Ex6 These two triangles are similar. Find x and y. Start by drawing the triangles out separately so that the corresponding angles match. x is a reduction of the side marked 3.6 so: Y is an enlargement of the side marked.5 so: Rs.f = = Es.f = = = So x = Scale Factor 3. 6 = x = 3cm So y = Scale Factor. 5 = y = 3cm Ex7 These two triangles are similar. Find x and y. Again, start by drawing the triangles out separately so that the corresponding angles match. Page 5 of 1

6 x is an enlargement of the side marked 3.8 so: 10 5 Es.f = = =. 5 4 So x = Scale Factor 38. = x = 9. 5cm y is a reduction of the side marked 8 so: 4 Rs.f = = = So y = Scale Factor 8 = y = 3. cm More Similar Triangles In a pair of similar triangles, the ratio of corresponding sides are equal: a b c = = d e f We say the corresponding sides are in proportion. This can be really useful when solving problems. Page 6 of 1

7 Ex8 Find the value of x given that the triangles are similar. Start by drawing out both triangles so that the corresponding angles match up. AB EB = AC DC 3 4 = x = 4( x + 3) 18 = 4x = 4x 6 = 4x x = 15. cm Page 7 of 1

8 Ex9 Find the value of x given that the triangles are similar. Start by drawing out both triangles so that the corresponding angles match up. PR QP = RT ST 18 x 3 = x 5 5( 18 x) = 3x 90 5x = 3x 90 = 3x + 5x 90 = 8x x = 11. 5cm Page 8 of 1

9 Other Similar Figures Any shapes that are similar, like triangles, will have their corresponding sides constant. Ex10 These quadrilaterals are similar. (a) Find the scale factor (b) Find the value of x Start by finding the scale factor: x is an enlargement of the side marked 4cm so: 3 Es.f = = 15. Note that we could have used any of the corresponding sides: 3.. = 5 = 3 75 = Then find the value of x: x x x = Scale Factor 4 = = 6cm Page 9 of 1

10 Ratio of Areas These two triangles are similar. The enlargement factor is k. 1 Area of smaller triangle = ab 1 1 Area of larger triangle = ka kb = abk The larger area is found by multiplying the smaller by k The scale factor for area is the scale factor for the length squared Ex11 Two sun cream bottles are similar with heights as shown. The larger bottle has a label of area 40cm. What is the area of the label on the smaller bottle? Page 10 of 1

11 Ex1 These two cuboids are similar. Calculate the surface area of the larger cuboid. S.A of smaller cuboid = ( 3) + ( 4) + ( 3 4) = 5cm Enlargement factor (length) = 8 4 = Enlargement factor for area = = 4 S.A of larger cuboid = 4 5 = 08cm Volume of Similar Solids Two cuboids are similar (with an enlargement factor of k.) Volume of smaller cuboid = abc cm Volume of larger cuboid = akbkck = abck cm Page 11 of 1

12 The scale factor for volume is the scale factor for the length cubed Ex13 A jug of milk has a height of 18cm holds 440ml of liquid. A smaller jug of milk in a similar shaped bottle has a height of 3cm. What volume of milk is in the smaller jug? Reduction factor (height) 3 1 = = Reduction factor (volume) = = Volume of the sample = 440 =. 04ml 16 3 Page 1 of 1

Warm-up. Translations Using arrow notation to write a rule. Example: 1) Write a rule that would move a point 3 units to the right and 5 units down.

Warm-up. Translations Using arrow notation to write a rule. Example: 1) Write a rule that would move a point 3 units to the right and 5 units down. Translations Using arrow notation to write a rule. Example: 1) Write a rule that would move a point 3 units to the right and 5 units down. (x, y) 2) Write a rule that would move a point 6 units down. (x,