1. Colegio Herma. Maths. Bilingual Departament

2. 2 of 67 If shapes are identical in shape and size then we say they are congruent. Congruent shapes can be mapped onto each other using translations, rotations and reflections. These triangles are congruent because A B C R P Q AB = PQ,BC = QR, and AC = PR.  A =  P,  B =  Q, and  C =  R.

3. 3 of 67 Two triangles are congruent if they satisfy the following conditions: Side, side side (SSS) 1) The three sides of one triangle are equal to the three sides of the other.

4. 4 of 67 2) Two sides and the included angle in one triangle are equal to two sides and the included angle in the other. Side, angle, side (SAS)

5. 5 of 67 Angle, angle, side (AAS) 3) Two angles and one side of one triangle are equal to the corresponding two angles and side in the other.

6. 6 of 67 4) The hypotenuse and one side of one right-angled triangle is equal to the hypotenuse and one side of another right-angled triangle. Right angle, hypotenuse, side (RHS) If they are right-angled triangles:

7. 7 of 67 If one shape is an enlargement of the other then we say the shapes are similar. The angle sizes in two similar shapes are the same and their corresponding side lengths are in the same ratio. A similar shape can be a reflection or a rotation of the original. These triangles are similar because  A =  P,  B =  Q, and  C =  R. A B C R P Q PQ AB = QR BC = PR AC A B C R P Q A B C R P Q A B C R P Q A B C R P Q A B C R P Q A B C R P Q

8. 8 of 67 A B C A’A’ B’B’ C’C’ a’ a b’b’ c’c’ c b

9. 9 of 67 Which of the following shapes are always similar? Any two squares? Any two rectangles? Any two isosceles triangles? Any two equilateral triangles? Any two circles? Any two cylinders? Any two cubes? Similar polygons Two polygons are similar if their corresponding angles are equals and their corresponding sides are proportional.

10. 10 of 67 We can find the scale factor for an enlargement by finding the ratio between any two corresponding lengths. Scale factor = length on enlargement corresponding length on original If a shape and its enlargement are drawn to scale, the 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.

11. 11 of 67 The scale factor for the enlargement is 9 / 6 = 1.5 The following rectangles are similar. What is the scale factor for the enlargement? 6 cm 9 cm

12. 12 of 67 The following shapes are similar. What is the size of each missing side and angle? 6 cm 53° a 5 cm 3 cm 4.8 cm 6 cm b The scale factor for the enlargement is 6 / 5 = 1.2 c d e 37° 53° 3.6 cm 7.2 cm 4 cm

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14. 14 of 67 THE USE OF THE THALES THEOREM 1. To divide a segment into equal parts: We have segment AB and we want to divide it into 11 equal parts. We do not know the length of segment AB, so we use the Thales’ Theorem to make the division. a)Draw a line from one of the ends of the segment, in the direction you want. b)b) As you have decided the line’s length, divide the line into 11 equal parts. c)c) Join the last point with the end of the segment. d)d) Draw parallel lines and the segment AB will be equally divided.

15. 15 of 67 2. We use shadows and proportionality to find the length of objects, height of buildings, persons, etc. The rays of sunlight form a triangle with the ground and the line of our object. The sunlight becomes the hypotenuse of a right- angled triangle

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17. 17 of 67 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.

18. 18 of 67 If we closed these two figures and form two triangles, we can guess they are similar, so their corresponding sides are proportional. a b c b á´ c´

19. 19 of 67 So, we can apply the proportinality rule for similar triangles: b´ a´ b a c´ c Remember that: The factor, k, is called the similarity ratio or scale factor if k > 1 the shape is enlarged, if k < 1 the shape is reduced.

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21. 21 of 67 Exercise A particular map shows a scale of 1 : 500. What is the actual distance if the map distance is 8 cm? Exercise 1 A particular map shows a scale of 1 cm : 5 km. What would the map distance (in cm) be if the actual distance is 14 km? Now do exercises 28 and 29 from page 209. Then do ex 72 to 75 from page 216.