01/11/20151 All of the ship would remain visible as apparent size diminishes. Whereas on a flat Earth

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01/11/20153 The lantern throws a shadow across the floor. What would happen if the lantern was closer to the man? What would happen if it was further away?

01/11/20154 (4, 6) (6, 5) (3, 2) (1, 5) (-4, 3) (-1, 1) (5, 3) (7, 1) (-7, 2) (-2, -1) (-4, -2) (4, -2) (2, -3) (-2, -4) y x – 7 – 6 – 5 – 4 – 3 – 2 – Starter Activity 2 What are the coordinates?

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6 Learning Objectives To enlarge the given image To find the centre of the enlargement Keywords: Enlargement, scale factor (SF), image, object, centre of enlargement (COE)

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8 Scale Factor (SF): the size by which an image is enlarged. SF can be a positive or negative, whole number or fractional number. Centre of Enlargement (COE): COE is the point from which the image is enlarged. If the COE is not given, we can draw the image anywhere.

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1001/11/2015Similar Shapes10 I want you to enlarge the rectangle. How many times bigger do you want it ? Twice as big. That’s a Scale Factor = 2 Do you mean twice the AREA or twice the line lengths ? You know exactly what I mean ! O.K. You mean twice the line lengths !

01/11/ /11/2015Similar Shapes11 What are the DIMENSIONS of the following enlargements ? S.F. = 2 S.F. = 3

01/11/ /11/2015Similar Shapes12 Copy the following shapes onto squared paper and sketch their enlargements. S.F. = 3 S.F.= 4 S.F.= 2 S.F. = 3 1) 2) 3) 4) 5)

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01/11/ /11/2015Similar Shapes15 If I wanted to enlarge a shape by say a Scale Factor = 2 I could draw it anywhere ! I will fix the position of the ‘image’ by using a Centre of Enlargement in the same way that the position of a lens fixes the position of an image. x C of E 4 Scale Factor = 2 x 2 = 8 x 87 x 2 = 14 x 4.47 x 2 = 8.94 x 7.28 x 2 = x All of the ‘light rays’ originate from and lead back to the Centre of Enlargement. If the enlargement is carried out on a grid then you may prefer to ‘count the squares.’

01/11/ /11/2015Similar Shapes16 x C of E Scale Factor = 2 xx xx If the enlargement is carried out on a grid then you may prefer to ‘count the squares.’ 4 left x 2 = 8 left 4 left and 2 down x 2 = 8 left and 4 down 7 left x 2 = 14 left 7 left and 2 down x 2 = 14 left and 4 down

01/11/ To enlarge the rectangle by scale factor x2 from the point shown. Centre of Enlargement Object A B C D Or Count Squares Image A/A/ B/B/ C/C/ D/D/ Enlargements from a Given Point 1. Draw the ray lines through vertices. 2. Mark off x2 distances along each line. 3. Draw and label image. X2

01/11/ D To enlarge the kite by scale factor x3 from the point shown. Centre of Enlargement Object A B C Or Count Squares Image A/A/ B/B/ C/C/ D/D/ Enlargements from a Given Point 1. Draw the ray lines through vertices. 2. Mark off x3 distances along lines from C of E. 3. Draw and label image. X3

01/11/ Enlargements from a Given Point 1. Draw the ray lines through vertices. 2. Mark off x4 distances along lines from C of E. 3. Draw and label image. To enlarge the triangle by scale factor x4 from the point shown. Centre of Enlargement Or Count Squares Object A B C Image C/C/ A/A/ B/B/ X4

01/11/ Negative Enlargements from a Given Point 1. Draw ray lines from each vertex through C of E. 2. Mark off x1distances along lines from C of E. 3. Draw and label image. B/B/ A/A/ C/C/ Image Image is Inverted D/D/ D Centre of Enlargement A B C Object To enlarge the kite by scale factor -1 from the point shown. Or Count Squares

01/11/ D Centre of Enlargement A B C Object To enlarge the kite by scale factor -2 from the point shown. Or Count Squares Negative Enlargements from a Given Point 2. Mark off distances x2 along lines from C of E. 3. Draw and label image. B/B/ A/A/ D/D/ C/C/ Image Image is Inverted 1. Draw ray lines from each vertex through C of E. -2

01/11/ Fractional Enlargements from a Given Point 1. Draw ray lines from C of E to each vertex. 2. Mark halfway distances along lines. 3. Draw and label image. B A D C Object Or Count Squares To enlarge the kite by scale factor x½ from the point shown. Centre of Enlargement A/A/ B/B/ D/D/ C/C/ Image ½

01/11/ Fractional Enlargements from a Given Point 1. Draw ray lines from C of E to each vertex. 2. Mark off 1/3 distances along lines. 3. Draw and label image. A’A’ B’B’ C’C’ D’D’ Image B A D C Or Count Squares To enlarge the rectangle by scale factor x1/3 from the point shown. Centre of Enlargement Object 1/3

01/11/ To enlarge the rectangle by scale factor x2 from the point shown. Centre of Enlargement Object A B C D Image A/A/ B/B/ C/C/ D/D/ Enlargements from a Given Point 1. Draw the ray lines through vertices. 2. Mark off x 2 distances along each line. 3. Draw and label image. No Grid 1

01/11/ D To enlarge the kite by scale factor x3 from the point shown. Centre of Enlargement Object A B C Image A/A/ B/B/ C/C/ D/D/ Enlargements from a Given Point 1. Draw the ray lines through vertices. 2. Mark off x3 distances along lines from C of E. 3. Draw and label image. No Grid 2

01/11/ Enlargements from a Given Point 1. Draw the ray lines through vertices. 2. Mark off x4 distances along lines from C of E. 3. Draw and label image. To enlarge the triangle by scale factor x4 from the point shown. Centre of Enlargement Object A B C Image C/C/ A/A/ B/B/ No Grid 3

01/11/ Negative Enlargements from a Given Point 1. Draw ray lines from each vertex through C of E. 2. Mark off x1distances along lines from C of E. 3. Draw and label image. B/B/ A/A/ C/C/ Image Image is Inverted D/D/ D Centre of Enlargement A B C Object To enlarge the kite by scale factor -1 from the point shown. No Grid 4

01/11/ D Centre of Enlargement A B C Object To enlarge the kite by scale factor -2 from the point shown. Negative Enlargements from a Given Point 2. Mark off distances x2 along lines from C of E. 3. Draw and label image. B/B/ A/A/ D/D/ C/C/ Image Image is Inverted 1. Draw ray lines from each vertex through C of E. No Grid 5

01/11/ x x Enlargement

01/11/ x 2 across 1 up Enlarge by a scale factor of 2 about the centre of enlargement x. 4 across 2 up

01/11/ Enlarge by a scale factor of ½ about the centre of enlargement x. x 6 down 7 across 3 down 3.5 across

01/11/ /11/2015Similar Shapes32 xx Scale Factor = 3 and centre of origin = x x 3 = 6.6 x x 3 = 4.2 x x 3 = 6.6 x or 1 right and 2 up x 3 = 3 right and 6 up x 1 right and 1 down x 3 = 3 right and 3 down x 2 right and 1 down x 3 = 6 right and 3 down x ImageObject

01/11/ /11/2015Similar Shapes33 Scale Factor = 2 centre of enlargement = x xx x 2 = 7.2 x x 2 = 6.4 x x 2 = 4.4 x x 2 = 2.8 x 3 left and 2 up x 2 = 6 left and 4 up x 3 left and 1 down x 2 = 6 left and 2 down x 1 right and 2 up x 2 = 2 right and 4 up x 1 right and 1 down x 2 = 2 right and 2 down x

01/11/ /11/2015Similar Shapes34 Enlarge the following shapes by the given Scale Factors. x S.F. = 2 1) X S.F. = 3 2) x S.F. = 2 3) X S.F. = 3 4)

01/11/ /11/2015Similar Shapes35 Fully describe each Enlargement : Give the Scale Factor and the coordinates of the Centre of Enlargement (Each square = 1 unit) x S.F. = 2 ( - 9, 7 ) y x 1) Object Image Object Image 2) S.F. = 3 x ( 3, 4 ) 3) Object Image S.F. = 4 x ( - 4, - 2 )

01/11/ To enlarge a shape on a centimetre grid, simply multiply the lengths by the scale factor. Hint: You only need to worry about the vertical and horizontal lengths, the diagonals will follow. 2cm 3cm 6cm 9cm Scale Factor = 3

01/11/ To enlarge a shape about a centre of enlargement, draw lines from the centre through the vertices. Scale Factor = 3 Now measure along the lines three times the original distance from the centre of enlargement to each vertex. This is where the corresponding vertex will appear. Tip: You could use compasses. Centre of Enlargement

01/11/ a a’ Scale Factor = 2 The original vertices should be labeled with normal letters. The corresponding vertices on the image should be labeled with dashed letters.

01/11/ Centre of Enlargement What if the CoE is inside the shape?

01/11/ What about if I need to find the centre of enlargement? x y We have found the centre of enlargement! (2, 1)

01/11/ Scale Factor = -1

01/11/ Enlarge this triangle by a scale factor of 3.

01/11/ Enlarge this triangle by a scale factor of X 3 3 6

01/11/ Enlarge this triangle by a scale factor of 3 using (2, 1) as the centre of enlargement. 2 1

01/11/ Enlarge this triangle by a scale factor of 3 using (2, 1) as the centre of enlargement. 2 1

01/11/ Enlarge this triangle by a scale factor of 3 using (2, 1) as the centre of enlargement Draw lines from the centre of enlargement through the vertices (corners) of the shape.

01/11/ Enlarge this triangle by a scale factor of 3 using (2, 1) as the centre of enlargement. 2 1 Draw lines from the centre of enlargement through the vertices (corners) of the shape. 2. Use the lines to find the corners of the enlarged shape

01/11/ Enlarge this triangle by a scale factor of 3 using (2, 1) as the centre of enlargement. 2 1 Draw lines from the centre of enlargement through the vertices (corners) of the shape. Use the lines to find the corners of the enlarged shape

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01/11/ Scale Factor = -1

01/11/ What about if I need to find the centre of enlargement? x y We have found the centre of enlargement! (2, 1)

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01/11/ /11/2015Similar Shapes53 Scale Factor = 1/2 x 10.2 ½ of 10.2 = 5.1 x 16.1 ½ of 16.1 = 8.05 x 10.2 ½ of 10.2 = 5.1 x 16.1 ½ of 16.1 = 8.05 x Scale factors less than 1 will produce images smaller than their objects. C of E Object Image You may prefer to count squares ! x Object 10 left and 2 up ½ of 10 left and ½ of 2 up = 5 left and 1 up x 10 left and 2 down ½ of 10 left and ½ of 2 down = 5 left and 1 down x 16 left and 2 up ½ of 16 left and ½ of 2 up = 8 left and 1 up x 16 left and 2 down ½ of 16 left and ½ of 2 down = 8 left and 1 down x Image

01/11/ /11/2015Similar Shapes54 Enlarge the following shapes by the given Scale Factors. x S.F.= 1/2 1) x 2) S.F.= 1/4 3) S.F.= 2/3 x S.F.= 1 5. x 4)

01/11/ /11/2015Similar Shapes55 Fully describe each Enlargement : Give the Scale Factor and the coordinates of the Centre of Enlargement (Each square = 1 unit) Object Image 1) S.F. = 1/2 C of E at ( - 6, 0 ) x x Object Image S.F. = 1/3 C of E at ( 3, 2 ) 2) Object Image S.F. = 1/4 x C of E at ( - 1, - 1 ) 3)

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01/11/ /11/2015Similar Shapes57 The lens in your eye produces an image using a Negative Scale Factor ! x Scale Factor = x -2 = x 2.83x -2 = x 4.47x -2 = x Object Image As with the other enlargements you could have carried out these negative enlargements by counting squares in the opposite directions.

01/11/ /11/2015Similar Shapes58 Enlarge the following shapes by the given Scale Factors. x S.F. = - 2 1) X S.F. = )

01/11/ /11/2015Similar Shapes59 Fully describe each Enlargement : Give the Scale Factor and the coordinates of the Centre of Enlargement (Each square = 1 unit) x Object Image 1) Enlargement, Scale Factor = - 3 Centre of Enlargement at ( - 7, 4 ) x Object Image Enlargement, Scale Factor = - 1/4 2) Centre of Enlargement at ( 5, - 1 )

01/11/ Fractional Enlargements from a Given Point 1. Draw ray lines from C of E to each vertex. 2. Mark halfway distances along lines. 3. Draw and label image. B A D C Object To enlarge the kite by scale factor x½ from the point shown. Centre of Enlargement A/A/ B/B/ D/D/ C/C/ Image No Grid 6

01/11/ Fractional Enlargements from a Given Point 1. Draw ray lines from C of E to each vertex. 2. Mark off 1/3 distances along lines. 3. Draw and label image. A’A’ B’B’ C’C’ D’D’ Image B A D C To enlarge the rectangle by scale factor x 1/3 from the point shown. Centre of Enlargement Object No Grid 7

01/11/ To enlarge the rectangle by scale factor x2 from the point shown. Centre of Enlargement Object A B C D Or Count Squares Enlargements from a Given Point Worksheet 1

01/11/ D To enlarge the kite by scale factor x3 from the point shown. Centre of Enlargement Object A B C Or Count Squares Enlargements from a Given Point Workshe et 2

01/11/ Enlargements from a Given Point To enlarge the triangle by scale factor x4 from the point shown. Centre of Enlargement Or Count Squares Object A B C Worksheet 3

01/11/ Negative Enlargements from a Given Point D Centre of Enlargement A B C Object To enlarge the kite by scale factor -1 from the point shown. Or Count Squares Worksheet 4

01/11/ D Centre of Enlargement A B C Object To enlarge the kite by scale factor -2 from the point shown. Or Count Squares Negative Enlargements from a Given Point Workshee t 5

01/11/ Fractional Enlargements from a Given Point B A D C Object Or Count Squares To enlarge the kite by scale factor x½ from the point shown. Centre of Enlargement Worksh eet 6

01/11/ Fractional Enlargements from a Given Point B A D C Or Count Squares To enlarge the rectangle by scale factor x 1/3 from the point shown. Centre of Enlargement Object Workshee t 7

01/11/ To enlarge the rectangle by scale factor x2 from the point shown. Centre of Enlargement Object A B C D Enlargements from a Given Point Worksheet 1A

01/11/ D To enlarge the kite by scale factor x3 from the point shown. Centre of Enlargement Object A B C Workshe et 2A

01/11/ Enlargements from a Given Point To enlarge the triangle by scale factor x4 from the point shown. Centre of Enlargement Object A B C Worksheet 3A

01/11/ Negative Enlargements from a Given Point D Centre of Enlargement A B C Object To enlarge the kite by scale factor -1 from the point shown. Worksheet 4A

01/11/ D Centre of Enlargement A B C Object To enlarge the kite by scale factor -2 from the point shown. Workshee t 5A Negative Enlargements from a Given Point

01/11/ B A D C Object To enlarge the kite by scale factor x½ from the point shown. Centre of Enlargement Worksh eet 6A Fractional Enlargements from a Given Point

01/11/ Fractional Enlargements from a Given Point B A D C To enlarge the kite by scale factor x 1/3 from the point shown. Centre of Enlargement Object Workshee t 7A

01/11/ The small rectangle has been enlarged as shown. Find the centre of enlargement. Object A C D Image A/A/ B/B/ C/C/ D/D/ Finding the Centre of Enlargement B Draw 2 ray lines through corresponding vertices to locate. Centre of Enlargement Find Centre 1

01/11/ B/B/ A/A/ D/D/ C/C/ Image D A B C Object The small kite has been enlarged as shown. Find the centre of enlargement. Finding the Centre of Enlargement Centre of Enlargement Draw 2 ray lines through corresponding vertices to locate. Find Centre 2

01/11/ Finding the Centre of Enlargement. B A D C Object The large kite has been enlarged by scale factor x ½ as shown. Find the centre of enlargement. A/A/ B/B/ D/D/ C/C/ Image Draw 2 ray lines through corresponding vertices to locate. Centre of Enlargement Find Centre 3

01/11/ The small rectangle has been enlarged as shown. Find the centre of enlargement. Object A C D Image A/A/ B/B/ C/C/ D/D/ Finding the Centre of Enlargement B Worksheet 8

01/11/ B/B/ A/A/ D/D/ C/C/ Image D A B C Object The small kite has been enlarged as shown. Find the centre of enlargement. Finding the Centre of Enlargement Worksheet 9

01/11/ Finding the Centre of Enlargement. B A D C Object The large kite has been enlarged by scale factor x ½ as shown. Find the centre of enlargement. B/B/ A/A/ D/D/ C/C/ Image Workshheet 10

01/11/ The small rectangle has been enlarged as shown. Find the centre of enlargement. Object A C D Image A/A/ B/B/ C/C/ D/D/ Finding the Centre of Enlargement B Worksheet 8A

01/11/ B/B/ A/A/ D/D/ C/C/ Image D A B C Object The small kite has been enlarged as shown. Find the centre of enlargement. Finding the Centre of Enlargement Worksheet 9A

01/11/ Finding the Centre of Enlargement. B A D C Object The large kite has been enlarged by scale factor x ½ as shown. Find the centre of enlargement. A/A/ B/B/ D/D/ C/C/ Image Workshheet 10A

01/11/ Enlargement A A’ Shape A’ is an enlargement of shape A. The length of each side in shape A’ is 2 × the length of each side in shape A. We say that shape A has been enlarged by scale factor 2.

01/11/ Enlargement When a shape is enlarged the ratios of any length in the image over the corresponding lengths in the original shape (the object) is equal to the scale factor. A B C A’ B’ C’ = B’C’ BC = A’C’ AC = the scale factor A’B’ AB 4 cm 6 cm 8 cm 9 cm 6 cm 12 cm 6 4 = 12 8 = 9 6 = 1.5

01/11/ Congruence and similarity Is an enlargement congruent to the original object? Remember, if two shapes are congruent they are the same shape and size. Corresponding lengths and angles are equal. In an enlarged shape the corresponding angles are the same but the lengths are different. The object and its image are similar. Reflections, rotations and translations produce images that are congruent to the original shape. Enlargements produce images that are similar to the original shape.

01/11/ Enlargement A A’ Shape A’ is an enlargement of shape A. The length of each side in shape A’ is 2 × the length of each side in shape A. We say that shape A has been enlarged by scale factor 2.

01/11/ Enlargement When a shape is enlarged the ratios of any length in the image over the corresponding lengths in the original shape (the object) is equal to the scale factor. A B C A’ B’ C’ = B’C’ BC = A’C’ AC = the scale factor A’B’ AB 4 cm 6 cm 8 cm 9 cm 6 cm 12 cm 6 4 = 12 8 = 9 6 = 1.5

01/11/ Find the scale factor A A’ Scale factor = 3 What is the scale factor for the following enlargements?

01/11/ Find the scale factor What is the scale factor for the following enlargements? Scale factor = 2 B B’

01/11/ Find the scale factor Scale factor = 3.5 C C’ What is the scale factor for the following enlargements?

01/11/ Find the scale factor Scale factor = 0.5 D D’ What is the scale factor for the following enlargements?

01/11/ Find the scale factor A A’ Scale factor = 3 What is the scale factor for the following enlargements?

01/11/ Find the scale factor What is the scale factor for the following enlargements? Scale factor = 2 B B’

01/11/ Find the scale factor Scale factor = 3.5 C C’ What is the scale factor for the following enlargements?

01/11/ Find the scale factor Scale factor = 0.5 D D’ What is the scale factor for the following enlargements?

01/11/ Scale factors between 0 and 1 What happens when the scale factor for an enlargement is between 1 and 0? When the scale factor is between 1 and 0, the enlargement will be smaller than the original object. Although there is a reduction in size, the transformation is still called an enlargement. For example, E E’ Scale factor = 2 3

01/11/ The centre of enlargement To define an enlargement we must be given a scale factor and a centre of enlargement. For example, enlarge triangle ABC by a scale factor of 2 from the centre of enlargement O. O A C B OA’ OA = OB’ OB = OC’ OC = 2 A’ C’ B’

01/11/ The centre of enlargement Enlarge quadrilateral ABCD by a scale factor of from the centre of enlargement O. O D A B C OA’ OA = OB’ OB = OC’ OC = OD’ OE A’ D’ B’ C’ 1 3 = 1 3

01/11/ This example shows the shape ABCD enlarged by a scale factor of –2 about the centre of enlargement O. O A B C D Negative scale factors When the scale factor is negative the enlargement is on the opposite side of the centre of enlargement. A’ D’ B’ C’

01/11/ Inverse enlargements An inverse enlargement maps the image that has been enlarged back onto the original object. What is the inverse of an enlargement of 0.2 from the point (1, 3)? In general, the inverse of an enlargement with a scale factor k is an enlargement with a scale factor from the same centre of enlargement. 1 k The inverse of an enlargement of 0.2 from the point (1, 3) is an enlargement of 5 from the point (1, 3).

01/11/ Enlargement on a coordinate grid

01/11/ Finding the centre of enlargement Find the centre the enlargement of A onto A’. A A’ This is the centre of enlargement Draw lines from any two vertices to their images. Extend the lines until they meet at a point.

01/11/ Finding the centre of enlargement Find the centre the enlargement of A onto A’. A A’ Draw lines from any two vertices to their images. When the enlargement is negative the centre of enlargement is at the point where the lines intersect.

01/11/ Describing enlargements

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01/11/ /11/ For shapes to be similar they must : 1)Have identical angles. 2)Have their sides in the same proportion. 4 2 = Two shapes are said to be SIMILAR when one is an ENLARGEMENT of the other.

01/11/ /11/ Match up the PAIRS of Similar Shapes. Rectangles NOT drawn to scale. 1) 2) 3) 4)5) 6) 7) 8) ÷ 3 = 2 8 ÷ 4 = 2 3 ÷ 2 = ÷ 10 = ÷ 2 = 5 15 ÷ 3 = 5 14 ÷ 4 = ÷ 2 = 3.5

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01/11/ /11/2015Similar Shapes111 For 2 Triangles to be Similar to each other you only need to check whether or not they have the same angles. If their angles are the same then their sides will automatically be in the same proportions Same angles so automatically Similar Triangles. Menu

01/11/ /11/2015Similar Shapes112 Match up the PAIRS of Similar Triangles. Triangles NOT drawn to scale. Menu 40° 80° 40° 60° 30° 1)2) 3) 4) 5) 6) 7) 8) 60° 70° 20° 60° 70° 60° 50°

01/11/ /11/2015Similar Shapes113 Menu

01/11/ /11/2015Similar Shapes cm 15 cm Menu x cm 30 cm Bob decides to enlarge a poster of himself. How wide will the enlargement be ? x 30 = × × 30 x = 10 × x = x = 20 cm

01/11/ /11/2015Similar Shapes115 Menu Bob’s work rival decides to reduce the poster so that it is only 3 cm wide. How long will it be ? x cm 3 cm 20 cm 30 cm x 3 = × × 3 x = 30 × 3 20 x = x = 4.5 cm

01/11/ /11/2015Similar Shapes116 Calculate the missing lengths. { Each pair of shapes are similar } Menu x x x x 14 x x ° 115° 1) 2) 3) 4) 5) 6)

01/11/ /11/2015Similar Shapes117 Menu

01/11/ /11/2015Similar Shapes118 Menu How high is the church spire ? 1)Hammer a stick into the ground. 2)Line up the top of the stick with the top of the spire. {You will need to put your eye to the ground} 3)We now have 2 Similar Triangles because … Parallel Common to both triangles Both are Right Angles Corresponding Angles

01/11/ /11/2015Similar Shapes119 Menu How high is the church spire ? 4)Measure the height of the stick. 2 m 5)Measure the distances from the ‘eye’ to the stick and the ‘eye’ to the church. 4 m 50 m 6)Let the height of the spire be called x. x 7)You may well find it easier seeing them as two separate triangles x 50 = ×× 50 x = 2 × 50 4 x = 25 m

01/11/ /11/2015Similar Shapes120 Menu Calculate the missing lengths x x x x x )2) 3) 4) Harder Problems

01/11/ /11/2015Similar Shapes121 Menu Calculate the missing lengths x > > > > > > x x x )2) 3) 4) AB C D E Prove that triangles ABC and CDE in question 3 are similar ACB = DCE (Vertically opposite angles) CDE = BAC (Alternate angles) CED = ABC (Alternate angles) ^^ ^^ ^^

01/11/ /11/2015Similar Shapes122 Menu

01/11/ /11/2015Similar Shapes123 Menu Each rectangle is enlarged using a Scale Factor = 2 Diagrams not drawn to scale     Work out the Areas of each of the rectangles S.F. = 2 1 × 2 = 22 × 4 = 8 2 × 3 = 64 × 6 = 24 5 × 6 = × 12 = × 10 = × 20 = 280 In each case how has the Area increased ? 2  8 6    280 × 4

01/11/ /11/2015Similar Shapes124 Menu Each rectangle is enlarged using a Scale Factor = 3 Diagrams not drawn to scale     Work out the Areas of each of the rectangles S.F. = 3 1 × 2 = 23 × 6 = 18 2 × 3 = 66 × 9 = 54 5 × 6 = × 18 = × 10 = × 30 = 630 In each case how has the Area increased ? 2  18 6    630 × 9

01/11/ /11/2015Similar Shapes125 Menu Each rectangle is enlarged using a Scale Factor = 4 Diagrams not drawn to scale     Work out the Areas of each of the rectangles S.F. = 4 1 × 2 = 24 × 8 = 32 2 × 3 = 68 × 12 = 96 5 × 6 = × 24 = × 10 = 7028 × 40 = 1120 In each case how has the Area increased ? 2  32 6    1120 × 16

01/11/ /11/2015Similar Shapes126 What is the connection between the Scale Factor and the increase in Area ? Menu Scale FactorIncrease in Area ( Area multiplier ) 2× 4 3× 9 4× 16 ( Scale Factor ) 2 = Increase in Area

01/11/ /11/2015Similar Shapes127 Menu Example 1 Area = 8 cm 2 Area = ? 5 cm 15 cm S.F. = 15 ÷ 5 = 3 ( Scale Factor ) 2 = Area multiplier 3 2 = 9 times New Area = 8 × 9 = 72 cm 2 Example 2 ( Scale Factor ) 2 = Area multiplier 4 cmx Area = 10 cm 2 Area = 250 cm 2 Area multiplier = 250 ÷ 10 = 25 times Scale Factor = Area multiplier  S.F. =  25 S.F. = 5 New base : 4 × 5 = 20 cm

01/11/ /11/2015Similar Shapes128 Menu Work out the following : {All of the shapes are Similar} Area = 6 cm 2 Area = ? 2 cm8 cm Area = ? Area = 45 cm 2 12 cm 4 cm Area = 180 m 2 Area = 20 m 2 x5 m 16 mx Area = 100 m 2 Area = 25 m 2 1)2) 3)4) 96 cm 2 5 cm 2 15 m 8 m

01/11/ /11/2015Similar Shapes129 Menu

01/11/ /11/2015Similar Shapes130 Menu Each CUBOID is enlarged using a Scale Factor = 2 Diagrams not drawn to scale. S.F. = 2     Work out the Volume of each cuboid. 1 × 1 × 2 = 2 2 × 2 × 4 = 16 1 × 2 × 3 = 6 2 × 4 × 6 = 48 2 × 2 × 3 = 12 4 × 4 × 6 = 96 2 × 3 × 4 = 24 4 × 6 × 8 = 192 In each case how has the Volume increased ? 2  16 6    192 × 8

01/11/ /11/2015Similar Shapes131 Menu Each CUBOID is enlarged using a Scale Factor = 3 Diagrams not drawn to scale. S.F. = 3     Work out the Volume of each cuboid. 1 × 1 × 2 = 2 3 × 3 × 6 = 54 1 × 2 × 3 = 6 3 × 6 × 9 = × 2 × 3 = 12 6 × 6 × 9 = × 3 × 4 = 24 6 × 9 × 12 = 648 In each case how has the Volume increased ? 2  54 6    648 × 27

01/11/ /11/2015Similar Shapes132 Menu Each CUBOID is enlarged using a Scale Factor = 4 Diagrams not drawn to scale. S.F. = 4     Work out the Volume of each cuboid. 1 × 1 × 2 = 2 4 × 4 × 8 = × 2 × 3 = 6 4 × 8 × 12 = × 2 × 3 = 12 8 × 8 × 12 = × 3 × 4 = 24 8 × 12 × 16 = 1536 In each case how has the Volume increased ? 2     1536 × 64

01/11/ /11/2015Similar Shapes133 What is the connection between the Scale Factor and the increase in Volume ? Menu Scale Factor Increase in Volume ( Volume multiplier ) 2× 8 3× 27 4× 64 ( Scale Factor ) 3 = Increase in Volume

01/11/ /11/2015Similar Shapes134 ( Scale Factor ) 3 = Volume multiplier Menu Example 1 Example 2 Volume = 50 m 3 Volume = ? 5 m 10 m Scale Factor = = 8 times New Volume = 50 × 8 = 400 m 3 ( Scale Factor ) 3 = Volume multiplier Volume = 20 m 3 Volume = 540 m 3 4 mx Volume multiplier = 540 ÷ 20 = 27 times Scale Factor =  Volume Multiplier 3 S.F. =  27 3 S.F. = 3 New width : 4 × 3 = 12 m

01/11/ /11/2015Similar Shapes135 Menu Work out the following : {All of the shapes are Similar} Volume = 15 m Volume = ? Volume = 3750 m 3 Volume = ? 420 Volume = 40 m 3 Volume = 1080 m 3 6 x x 36 Volume = 200 m 3 Volume = m 3 1)2) 3)4) 120 m 3 30 m 3 18 m 9 m

01/11/ /11/2015Similar Shapes136 Menu

01/11/ /11/2015Similar Shapes137 Menu Shapes are said to be CONGRUENT when they have the same angles and their sides are the same length. They are identical. They would fit perfectly over each other. 80° 100° 70° 110° 70°80° 110°100° 110° 80° 70°

01/11/ /11/2015Similar Shapes138 Menu For Triangles to be CONGRUENT ( hence identical ) they have to fulfil one of four conditions : 1)If their sides are all the same length then the triangles are identical ( Congruent ). Side Side Side S.S.S

01/11/ /11/2015Similar Shapes139 Menu For Triangles to be CONGRUENT ( hence identical ) they have to fulfil one of four conditions : 2)If 2 of their sides are the same length and their INCLUDED angles are the same then the triangles are identical ( Congruent ). Side Angle Side S.A.S 40° The INCLUDED angle lies between the 2 pairs of equal length sides. 40°

01/11/ /11/2015Similar Shapes140 Menu For Triangles to be CONGRUENT ( hence identical ) they have to fulfil one of four conditions : 3)If 2 of their angles are the same and also 1 of their corresponding sides are the same then the triangles are identical ( Congruent ). Angle Angle Side A.A.S 40° 70° A corresponding side lies opposite to one of the identical angles. OR

01/11/ /11/2015Similar Shapes141 For Triangles to be CONGRUENT ( hence identical ) they have to fulfil one of four conditions : 4)If they both have Right angles, they both have the same Hypotenuse and one other side is the same length then the triangles are identical ( Congruent ). Right angle Hypotenuse Side R.H.S Menu OR

01/11/ /11/2015Similar Shapes142 Menu Summary of conditions for Congruent Triangles. Side Side Side S.S.SSide Angle Side S.A.S Angle Angle Side A.A.S Hypotenuse Right angle Hypotenuse Side R.H.S

01/11/ /11/2015Similar Shapes143 Menu Which triangles are Congruent to the RED triangle. You must give reasons. ie SSS, AAS, SAS, RHS Triangles not drawn to scale ° 60°40° 60°40° ° ° °40° 9.8 1) 2)3) 4) 5) A.A.S S.S.S 6) 80° 60° 40° S.A.S R.H.S problems

01/11/ /11/2015Similar Shapes144 Which triangles are Congruent to the RED triangle. You must give reasons. ie SSS, AAS, SAS, RHS Menu Triangles not drawn to scale ° 5 37° A.A.S 5 3 R.H.S 53° ° ° 5 3 S.A.S or R.H.S or A.A.S 37°

01/11/ Enlargement Transforms a shape using –A Centre of Enlargement –A Scale Factor Exam questions sometimes involve enlargements on an x,y grid

01/11/ Draw a 3 times enlargement of this shape DIAGONAL lines are best drawn LAST A shape we change is called an OBJECT This OBJECT was enlarged with a SCALE FACTOR of 3 The shape it becomes is called the IMAGE

01/11/ USING A CENTRE OF ENLARGEMENT Enlarge Triangle by scale factor 2 With Centre of Enlargement (0,0) x Move one VERTEX at a time

01/11/ After Scale factor 2 Enlargement, how many Times will OBJECT fit inside its IMAGE?