Congruence and similarity
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The document explains the concepts of congruence and similarity in shapes, emphasizing that congruent shapes are identical in size and shape while similar shapes maintain proportional dimensions with congruent angles. It includes methods for finding scale factors for enlargements and offers practical examples, such as dimensions in architectural plans and measuring heights using shadows. Additionally, it presents various geometric scenarios involving congruent and similar triangles.
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Congruence and similarity
- 1. © Boardworks Ltd 20131 of 22 This icon indicates the slide contains activities created in Flash. These activities are not editable. For more detailed instructions, see the Getting Started presentation. This icon indicates an accompanying worksheet. This icon indicates teacher’s notes in the Notes field.
- 2. © Boardworks Ltd 20132 of 22
- 3. © Boardworks Ltd 20133 of 22 Congruence If shapes are identical in shape and size then they are congruent. Congruent shapes can be mapped onto each other using translations, rotations and reflections. What makes these triangles congruent? AB = PQ, BC = QR, and AC = PR. A = P, B = Q, and C = R. C A B P Q R
- 4. © Boardworks Ltd 20134 of 22 Congruent triangles
- 5. © Boardworks Ltd 20135 of 22 Congruent triangles
- 6. © Boardworks Ltd 20136 of 22
- 7. © Boardworks Ltd 20137 of 22 What makes these triangles similar? Similar shapes If one shape is an enlargement of the other then the shapes are similar. The angles in two similar shapes are the same size and the lengths of their corresponding sides are in the same ratio. A similar shape can be a reflection or a rotation of the original. A = P, B = Q, and C = R. PQ AB = QR BC = PR AC A B C R P Q A P B C R Q
- 8. © Boardworks Ltd 20138 of 22 Which of the following shapes are always similar? Similar shapes any two squares any two rectangles any two isosceles triangles any two cubes any two cylinders any two circles any two equilateral triangles
- 9. © Boardworks Ltd 20139 of 22 Similar triangles 1
- 10. © Boardworks Ltd 201310 of 22 Similar triangles 2
- 11. © Boardworks Ltd 201311 of 22
- 12. © Boardworks Ltd 201312 of 22 Scale factor = length on enlargement corresponding length on original Finding the scale factor of an enlargement The scale factor for an enlargement is found by the ratio between any two corresponding lengths. The two corresponding lengths can be found using a ruler. Always make sure that the two lengths are written using the same units before dividing them.
- 13. © Boardworks Ltd 201313 of 22 The scale factor for the enlargement is: 9 ÷ 6 = 1.5 Finding the scale factor of an enlargement The following rectangles are similar. 6cm 9cm What is the scale factor for the enlargement?
- 14. © Boardworks Ltd 201314 of 22 Finding the lengths of missing sides 6cm 53° a 5cm 3cm 4.8cm 6cm b c d e 37° 37° 53° 3.6cm 7.2cm 4cm The following shapes are similar. What is the size of each missing side and angle? The scale factor for the enlargement is 6 ÷ 5 = 1.2
- 15. © Boardworks Ltd 201315 of 22 Finding lengths in two similar shapes
- 16. © Boardworks Ltd 201316 of 22
- 17. © Boardworks Ltd 201317 of 22 Architect's plans An architect creates a model of a house at one-fortieth of the actual size. Complete the table of some of the dimensions. model length model width model height real length real width garage porch chimney roof 1 real height 6m 4m10cm 2m15cm 5cm 2.6m 2.2m5.5cm 2m6.5cm 5cm 0.4m 0.48m1.2cm 1m1cm 2.5cm 8m 6.4m16cm 1.2m20cm 3cm
- 18. © Boardworks Ltd 201318 of 22 Paper sizes What are the dimensions of A7 and A8?
- 19. © Boardworks Ltd 201319 of 22 Bridge structures and congruency Identify the congruent shapes in these bridge designs.
- 20. © Boardworks Ltd 201320 of 22 Satellite photography Object size Image size focal length of camera height above Earth The camera on the satellite takes a photo of something on Earth. By using similar triangles can you work out the length of the object? You know the height above ground (150km), the image size (25mm) and the focal length (100mm).
- 21. © Boardworks Ltd 201321 of 22 Using shadows to measure height In ancient times, surveyors measured the height of tall objects by using a stick and comparing the length of its shadow to the length of the shadow of the tall object.
- 22. © Boardworks Ltd 201322 of 22 Using shadows to measure height