Believethatyoucandoitandyouar. ngascannotdoonlynotyetbelieve. Mathematics. thatyoucandoitandyouarehalfw. Stage 3

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1 Believethatyoucandoitandyouar ehalfwaytherethereisnosuchthi ngascannotdoonlynotyetbelieve Mathematics thatyoucandoitandyouarehalfw Stage 3 aytherethereisnosuchthingasca Shape & Space nnotdoonlynotyetbelievethatyo S J Cooper ucandoitandyouarehalfwayther ethereisnosuchthingascannotd oonlynotyetbelievethatyoucand oitandyouarehalfwaytherether eisnosuchthingascannotdoonlyn otyetbelievethatyoucandoitand youarehalfwaytherethereisnos uchthingascannotdoonlynotyet Believethatyoucandoitandyouar ehalfwaytherethereisnosuchthi

2 This book is on loan to

3 Geometry (3) Construction of triangles Remember do not remove any construction lines or arcs. 1. Draw a triangle ABC whose sides are AB = 7 cm, AC = 5 cm and BC = 4 cm. Measure and write down the size of angle A. 2. Draw a triangle LMN whose sides are LM = 9 cm, MN = 4 cm and LN = 7 cm. Measure and write down the size of angle N. 3. Draw a triangle PQR whose sides are PQ = 4 cm, PR = 4 cm and QR = 6 cm. Measure and write down the size of angle Q. 4. Draw a triangle DEF whose sides are DE = 8 cm, EF = 5 cm and DF = 7 cm. Measure and write down the size of angle D. 5. Draw a triangle ABC whose sides are AB = 5.4 cm, AC = 3.7 cm and BC = 6.3 cm. Measure and write down the size of angle C. 6. Draw a triangle XYZ whose sides are XY = 7.2 cm, XZ = 6.2 cm and YZ = 9.4 cm. Measure and write down the size of angle Z. 7. Draw a triangle LMN whose sides are LM = 4.8 cm, MN = 5.5 cm and LN = 11 cm. Measure and write down the size of angle N. 8. With the aid of compasses, protractor, rulers, etc... Draw accurately the following triangles and find the lengths required. (a) (b) Q (c) M 4 cm 4 cm 10.4 cm 7.2 cm 13.4 cm 3 cm N E 2 cm R 9.1 cm 6.2 cm P L Angle E =? Angle P =? Angle M =?

4 Geometry (4) Construction of triangles REMEMBER DO NOT REMOVE ANY CONSTRUCTION LINES OR ARCS. Use Pencil, Ruler, Compass and protractor for questions 1 to Draw a triangle ABC whose side AB = 7 cm and angles BAC = 40 and ABC = 50 Measure and write down the length of side BC. 2. Draw a triangle XYZ, where XY = 4 cm, ZXY = 70 and ZYX = 70. Measure and write down the length of ZX. 3. Draw a triangle DEF where DE = 6 cm, EDF = 54 and DEF = 31. Measure and write down the length of EF. 4. Draw a triangle PQR where PQ = 4.2 cm, PQR = 35 and QPR = 117. Measure and write down the length of side QR. 5. Draw an accurate drawing of the triangle opposite. Use Pencil, Ruler and Compass only for questions 6 to Draw a triangle PQR where PQ = 3 cm, P and Qˆ R = 90 Measure and write down the length of side QR. Q PˆR = cm Draw a triangle BCD where BC = 8.4 cm, Measure and write down the length of side BD. B ĈD =15 and C Bˆ D = Draw a triangle HIJ where HI = 6.7 cm, Measure and write down the length of side IJ. I ĤJ = 75 and H ÎJ = Draw a triangle ABC where AB = 5 cm, Measure and write down the length of side AC. A Bˆ C =120 and B ÂC = Draw a triangle DEF where DE = 7.3 cm, Measure and write down the length of side DF. E F = 45 and Dˆ D ÊF = 30.

5 Geometry (5) Construction of triangles REMEMBER DO NOT REMOVE ANY CONSTRUCTION LINES OR ARCS. Use pencil, ruler, compass and protractor for questions 1 to Draw a triangle LMN where LM = 4 cm, Measure and write down the length of LN. LMN= ˆ 30 and MN = 5 cm. 2. Draw the triangle PQR where PQ = 7 cm, Measure and write down the size of P Qˆ R. QPR ˆ = 70 and PR = 4 cm. 3. Draw a triangle JKL where JK = 5 cm, Measure and write down the length of JL. JKL ˆ = 55 and KL = 5.6 cm. 4. Draw the triangle XYZ where XY = 3.8 cm, Measure and write down the size of Xˆ YZ. Yˆ XZ =125 and XZ = 5.1 cm. 5. Draw a triangle ABC where AB = 4.3 cm, ABC = 45 and BC = 5.6 cm. Measure and write down the length of AC. Use pencil, ruler and compass only for questions 6 to Draw the triangle DEF where DF = 5.3 cm, Measure and write down the size of FDE. D FˆE = 60 and FE = 6.4 cm. 7. Draw a triangle STU where ST = 10.7 cm, Measure and write down the length of TU. T ŜU = 45 and SU = 8.5 cm. 8. Draw the triangle EFG where EF = 9.4 cm, Measure and write down the size of FG ˆ E. E FˆG = 30 and FG = 6.7 cm. 9. Draw the triangle PQR where PQ = 8.4 cm, P R =135 and QR = 4.7 cm. Qˆ Measure and write down the size of FG ˆ E. 10.Draw the triangle EFG where EF = 3.4 cm, Measure and write down the size of FG ˆ E. E FˆG =15 and FG = 5.5 cm.

6 11. Construct rectangle ABCD where AB = 9 cm and BC = 4 cm. State the length of the diagonal AC. 12. Construct a rectangle which has dimensions 5.3cm by 11.7cm. 13. Construct rectangle LMNO where LM = 3.9 cm and MN = 6.8 cm. State the length of the diagonal MO. 14. Construct a rectangle which has dimensions 10.3cm by 5.2cm.

7 Geometry (6) Types of Polygons 1. Name each of the following types of triangles (a) (b) (c) (d) (e) (f) 2. Draw a set of axes from 5 to 5 for each of the following problems. Plot the coordinates for each of the following. Join up the points to form the quadrilateral ABCD. What name is given to each shape drawn? (i) A( 2, 2), B(2, 4), C(2, 1), D( 2, 1) (ii) A( 2, 3), B(, 4), C(1, 2), D( 1, 1) (iii) A(4, 4), B( 2, 2), C(0, 0), D(3, 1) (iv) A(2, 3), B(0, 4), C( 3, 2), D(3, 1) (v) A( 1, 1), B( 2, 2), C(2, 3), D(3, 0) 3. (a) What name best describes a parallelogram with all angles at 90? (b) What name best describes a parallelogram with all sides equal in length? (c) What name best describes a parallelogram with all sides equal in length and all angles at 90? 4. Name each of the following quadrilaterals (a) BCDN (b) JMHI A B C (c) JMLK K L D N (d) DEFG (e) DGHN (f) LKAB (g) LBNH J M G I H E F

8 Geometry (7) Solids 1. For each of the tabulated solids below count the number of faces, vertices (corners) and edges. Enter the numbers in the appropriate place. In the last column work out the value of F + V E for each line. State what you notice. Cube Cuboid Square based pyramid Tetrahedron Triangular prism Number of Faces (F) Number of Vertices (V) Number of Edges (E) F + V E 2. A block of butter is in the shape of a cuboid until someone cuts away a corner with a knife, as shown. Count up faces, vertices and edges on the remainder of the butter shown. Complete the following. F =.. V =. E =. F + V E =. 3. (a) Using a pencil draw a sketch of a square based pyramid. Now take away the top corner using a rubber and redraw it to look as though someone had cut it away. (b) Complete the following for the remainder of the shape. F =.. V =. E =. F + V E =. 4. Here are some views of geometrical solids of the type drawn in class. State which they could be. [Some will have more than one answer!] (i) (ii) (iii) (iv)

9 5. This is a cuboid (edges not equal in length) and shows a plane of symmetry. i) Use tracing paper to copy the outline and dotted (hidden) lines into your exercise book. On your diagram draw a different plane of symmetry. ii) Repeat the exercise in (i) and draw another different plane of symmetry. 6. This is a cube (all edges equal). It will have three planes of symmetry similar to the cuboid in question 1. Shown is another plane of symmetry. i) Use tracing paper to copy the outline and dotted lines into your exercise book. Draw a new plane of symmetry. ii) Repeat the exercise of (i) as many times as you need to until all planes of symmetry have been found. iii) How many planes of symmetry does the cube have? 7. This is a square based pyramid and shows a plane of symmetry. i) Use tracing paper to copy the outline and dotted lines into your exercise book. Draw a new plane of symmetry. ii) Repeat the exercise of (i) as many times as you can have until all planes of symmetry have been found. 8. This is a sphere with a plane of symmetry. Draw a sphere into your book along with another plane of symmetry. How many planes of symmetry could be drawn?

10 9. This is a cylinder with a plane of symmetry. Draw a cylinder into your book with a different plane of symmetry. How many planes of symmetry could be drawn? 10. (i) Using a square (side 2cm) complete a net for a square based pyramid each edge of which will be length 4 cm. (ii) Draw on card a net for a square based pyramid of length 4cm. Add suitable flaps, cut out your net and glue together. 11. This is a sketch of a net for a regular tetrahedron, the dotted lines indicating folds. Construct on card an equilateral triangle of side 8 cm and mark the midpoints. Join the midpoints with dotted lines. Draw some flaps. Cut out your net; Use a pritt stick to glue together in the form of a regular tetrahedron, each edge of which should be of length 4 cm.

11 Geometry(8) Angles in a straight line Work out the lettered angles for each of the following diagrams. Remember to show your working. All diagrams are not drawn to scale

12 Geometry(9) Angles at a point Work out the lettered angles for each of the following diagrams. Remember to show your working. All diagrams are not drawn to scale

13 Geometry(10) Opposite Angles Work out the lettered angles for each of the following diagrams. Remember to show your working. All diagrams are not drawn to scale

14 Geometry(11) Corresponding Angles Work out the lettered angles for each of the following diagrams. Remember to show your working. All diagrams are not drawn to scale

15 Geometry(12) Alternate Angles Work out the lettered angles for each of the following diagrams. Remember to show your working. All diagrams are not drawn to scale

16 Geometry(13) Angles in a triangle Work out the missing angles in each of the following triangles. Remember to show your working. All diagrams are not drawn to scale Two sides of a triangle measured 83 and 51, what is the size of the third side? 14. Kath measured the angles of a triangle as 62, 87 and 31. Are the measurements likely to be correct on this evidence?

17 Geometry(14) Angles in a Polygon 1. Work out the size of the lettered interior angles for each of the following regular polygons (a) (b) (c) 2. Calculate the missing angles in the following parallelograms (a) (b) (c) 3. The diagram below shows two regular polygons joined together. Find the value of the angled labelled x. 4. Find the interior angle of a regular polygon which has 7 sides. Give your answer correct to 1 decimal place. 5. How many sides has a regular polygon if its interior angle is 168

18 6. The diagram below is of a regular octagon. Work out the size of the angle marked x. 7. The diagram below is of a regular hexagon. Work out the size of the angle marked y. 8. Explain why the sum of the interior angles of a regular heptagon is 900. The diagram below might help with your explanation. 9. A regular polygon has 20 sides. Work out the size of each interior angle. 10. The diagram below shows part of a regular polygon. How many sides would this polygon have?

19 Geometry(15) Area & perimeter Work out the area and perimeter for each of the following rectangles cm 12 cm 9 cm 4 cm 11cm 8cm cm 21cm 14cm 15cm 5cm 3cm m 16m 12m 15m 7m 13m mm 23m 6m 40mm 6m 35m 13. The rectangle below has an area of 210 cm 2. Work out the length of this rectangle. 3cm

20 14. Given that the perimeter of the rectangle below is 44cm, work out the length of the rectangle. 8 cm 15. Work out the area for each of the following triangles. (a) (b) (c) 7 cm 9 cm 12cm 4 cm 8 cm 7cm (d) (e) (f) 5m 14m 8cm 17cm 6cm 20cm (g) (h) (i) 16. Given that the area of the triangle below is 28 cm 2, work out the base length.

21 17. A triangle has base length 10 cm and area 50 cm 2. Work out the height of the triangle. 18. Which of the shapes below has the greatest area? 19. Copy and complete the table below Base Length Height Area (a) 7 cm 14 cm (b) 6 cm 39 cm 2 (c) 11 cm 44 cm 2 (d) 9cm 72cm 2 (e) 5cm 15cm 20. For the triangle below Sharon says that the area is equal to 130cm 2. Explain why she is wrong.

22 21. Work out the shaded area for each of the following: (a) (b) 15 (c) Work out the shaded area for each of the following: (a) 15 cm (b) 22 cm (c) 15 cm 3 cm 11 cm 17 cm 9 cm 22 cm 34 cm 18 cm 8 cm 31 cm (d) 7 cm (e) 18 cm 4 cm 4 cm 13 cm 4 cm 4 cm 6 cm 10 cm 13 cm 7 cm 2 cm 8 cm 12 cm 23. Work out the area for each of the following (all measurement are in cm): (a) (b) (c)

23 Geometry(16) Area of a trapezium 1. Calculate the area for each of the following (a) (b) (c) (d) (e) (f) 2. Given that the area of the trapezium below is 56 cm 2, find its height. 3. The area of the trapezium below is 36cm 2. Find the value of x.

24 Geometry(17) Volume of a prism 4. Without a calculator find (a) the base area (b) the volume for the following cuboids (a) (b) (c) (d) (e) (f) (g) (h) (i) 5. A beam is 5 metres long, 6 metres wide and 4 metres high. Find how many cubic metres of concrete was used to make the beam. 6. A classroom has a volume of 480m 3, if the length and width of the room are 8m and 7.5m respectively, how high is this classroom? 7. A cube has side 6 cm. Work out the volume of this cube. 8. A cuboid has volume 630cm 3.Given that two of the dimensions are 10 and 7cm, what is the length of the third dimension? 9. A cuboid has volume 240 cm 3. Work out a set of possible dimensions for this cuboid.

25 10. Without a calculator find the volume for each of the following triangular based prisms. (a) (b) (c) (d) (e) (f) 11. A triangular based prism has end area 30 cm 2, and volume 210cm 3. How long is the prism? 12. Given that the volume of the solid below is 640 cm 3, find its height. 13. Which of the solids below has the greatest volume? Show your working.

26 Geometry(18) Surface Area 1. Work out the surface area for each of the following cubiods (a) (b) (c) (d) (e) (f) (g) (h) (i) 2. A beam is 5 metres long, 6 metres wide and 4 metres high. Find how many cubic metres of concrete was used to make the beam. 3. The diagram opposite is of an open box. Work out its surface area.

27 Geometry(19) Transformations: Reflections Copy and complete each of the following transformations 1. Reflect the shape below in the x-axis 2. Reflect ABC in the y-axis. 3. Reflect the shape below in the y-axis 4. Reflect LMN in the x-axis.

28 5. Reflect ABCD in the line x = 1 6. Reflect DEF in the line y = 2 7. Reflect EFG in the line x = 1 8. Reflect PQR in the line y = 2

29 9. Reflect KLMN in the line y = Reflect HIJ in the line y = Reflect STU in the line x = Reflect ABC in the line y = 1

30 Geometry(20) Transformations: Rotations Copy and complete each of the following transformations 1. Rotate the object 90 clockwise 2. Rotate ABC 90 anticlockwise. with centre (0,0) with centre (1,2) 3. Rotate the object 180 with centre (3,0) 4. Rotate ABC 90 clockwise. with centre ( 1, 1)

31 5. Rotate the object 90 anticlockwise 6. Rotate DEF 180 with centre (1, 2) with centre (0,1) 7. Rotate the object 90 clockwise 8. Rotate PQR 90 anticlockwise. with centre (0, 1) with centre ( 1,0)

32 9. Rotate the object 180 with centre (2,0) 10. Rotate HIJ 90 clockwise. with centre (1,2) 11. Rotate the object 90 anticlockwise 12. Rotate ABC 180 with centre (1,1) with centre (1, 3)

33 Geometry(21)Transformations: Translations 1. Describe the transformation which moves Triangle PQR onto triangle STU 2. Draw each of the following transformations for the triangle below (a) Translation of (b) Translation of 2. Label the image A 4 1. Label the image B 2 (c) Translation of (d) Translation of 4. Label the image C 1 2. Label the image D 5

34 3. Describe the transformation for each of the following a) Triangle PQR onto Triangle ABC b) Triangle LMN onto Triangle ABC c) Triangle ABC onto Triangle DEF d) Triangle DEF onto Triangle LMN 4. Draw each of the following transformations for the triangle below 3 a) Translation of. Label the image A 6 1 b) Translation of. Label the image B 2 4 c) Translation of. Label the image C 1 2 d) Translation of. Label the image D 5

35 Geometry(22) Enlargements 1. Describe the transformation below which places ABC onto DEF. 2. Enlarge LMN by a scale factor of 2 centre (0, 0); Label the image L 1 M 1 N 1

36 3. Enlarge ABC by a scale factor of 4 centre (3, 2). Label the image A 1 B 1 C 1 4. Enlarge the object below with centre (-1, 1) by a scale factor 3.

37 5. Enlarge the object by a scale factor of 3 centre of enlargement (2, 1) 6. (a) Plot the points A(1, 1), B(1, 4) and C(3, 4) and join up the points to form a triangle ABC. (b) Enlarge the triangle ABC by a scale factor of 2 centre (1, 3) 7. (a) Plot the points P(0, 2), Q( 2, 2) and R( 1, 0) and join up the points to form a triangle PQR. (b) Enlarge the triangle PQR by a scale factor of 3 centre ( 1, 1) 8. (a) Plot the points L(1,0), M(0, 2) and N(4, 2) and join up the points to form a triangle LMN. (c) Enlarge the triangle LMN by a scale factor of 2 centre (2, 0) 9. (a) Plot the points D(1, 1), E(1, 2) and F(3, 0) and join up the points to form a triangle DEF. (d) Enlarge the triangle DEF by a scale factor of 3 centre (2, 2)

38 Geometry (23) Pythagoras Theorem 1. Work out the length of the hypotenuse for each of the following, giving your answers correct to 1 decimal place. [all measurements are in centimetres] (a) (b) 7 (c) 2 a 6 7 b c (d) (e) (f) d e 3 9 f (g) 2 (h) 10 (i) (j) 14 g 26 h 2.3 i 4.7 j 18 6 Work out the required lengths for each of the following, giving your answers to 2 decimal places. 2. Find a 3. Find b 16cm 6cm a 5cm b 14cm 4. Find AC A 5. Find EF E 11cm D 10cm 17cm C 18cm B 6. Find PR 7. Find LM R P 8cm 8cm Q M 8.3m F L 5.1m N

39 8. Find p 9. Find x 3.7cm p 19 cm 12.1cm A x 23cm 10. Find AC 9.9m C 13.1m B 10. A ladder is resting against a wall. The ladder is 2.1 metres from the wall and reaches 3.6 metres above the ground. Work out the length of the ladder. 11. A support is attached at a height of 3m and is fixed to the ground at a distance of 1.2m from the base. Calculate the length of the support.