mathematicsbtrcc.pbworks.com/w/file/fetch/112212487/stage 3 geometry...geometry (4) construction of...

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Mathematics

Stage 3

Shape & Space

S J Cooper

This book is on loan to ……………………………………………………………………………………………………………………….

……………………………………………………………………………………………………………………….

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) (c)

Angle E = ? Angle P = ? Angle M = ?

6.2 cm

7.2 cm 10.4 cm 4 cm

13.4 cm

9.1 cm

REMEMBER DO NOT REMOVE ANY CONSTRUCTION LINES OR ARCS.

Use Pencil, Ruler, Compass and protractor for questions 1 to 5.

1. 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 10.

6. Draw a triangle PQR where PQ = 3 cm , 90 = RQP and 30 = RPQ .

Measure and write down the length of side QR.

7. Draw a triangle BCD where BC = 8.4 cm , 15 = DCB and 60 = DBC .

Measure and write down the length of side BD.

8. Draw a triangle HIJ where HI = 6.7 cm , 75 = JHI and 60 = JIH .

Measure and write down the length of side IJ.

9. Draw a triangle ABC where AB = 5 cm , 120 = CBA and 30 = CAB .

Measure and write down the length of side AC.

10. Draw a triangle DEF where DE = 7.3 cm , 45 = FDE and 30 = FED .

Measure and write down the length of side DF.

4.6 cm

24 107

REMEMBER DO NOT REMOVE ANY CONSTRUCTION LINES OR ARCS.

Use pencil, ruler, compass and protractor for questions 1 to 5.

1. Draw a triangle LMN where LM = 4 cm , 30 = NML ˆ and MN = 5 cm.

Measure and write down the length of LN.

2. Draw the triangle PQR where PQ = 7 cm, 70 = RPQ ˆ and PR = 4 cm.

Measure and write down the size of RQP .

3. Draw a triangle JKL where JK = 5 cm , 55 = LKJ ˆ and KL = 5.6 cm.

Measure and write down the length of JL.

4. Draw the triangle XYZ where XY = 3.8 cm, 125 = ZXY ˆ and XZ = 5.1 cm.

Measure and write down the size of ZYX ˆ .

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 13.

6. Draw the triangle DEF where DF = 5.3 cm, 60 = EFD and FE = 6.4 cm.

Measure and write down the size of FDE .

7. Draw a triangle STU where ST = 10.7 cm , 45 = UST and SU = 8.5 cm.

Measure and write down the length of TU.

8. Draw the triangle EFG where EF = 9.4 cm, 30 =G FE and FG = 6.7 cm.

Measure and write down the size of EGF ˆ .

9. Draw the triangle PQR where PQ = 8.4 cm, 135 = RQP and QR = 4.7 cm.

10.Draw the triangle EFG where EF = 3.4 cm, 15 =G FE and FG = 5.5 cm.

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.

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

4. Name each of the following quadrilaterals

(a) BCDN

(b) JMHI

(c) JMLK

(d) DEFG

(e) DGHN

(f) LKAB

(g) LBNH

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.

Number of

Faces (F)

Number of

Vertices (V)

Number of

Edges (E)

F + V – E

Cuboid

Square based pyramid

Tetrahedron

Triangular prism

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)

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?

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.

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.

1. 2. 3.

4. 5. 6.

7. 8. 9.

10. 11. 12.

13. 14. 15.

Geometry(9) Angles at a point

1. 2. 3.

4. 5. 6

7. 8. 9.

Geometry(10) Opposite Angles

1. 2. 3.

4. 5. 6.

7. 8. 9.

Geometry(11) Corresponding Angles

1. 2. 3.

4. 5. 6.

7. 8. 9.

10. 11. 12.

Geometry(12) Alternate Angles

1. 2. 3.

4. 5. 6.

7. 8. 9.

10. 11. 12.

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

1. 2. 3.

4. 5. 6.

7. 8. 9.

10. 11. 12.

13. 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?

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°

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?

Geometry(15) Area & perimeter

Work out the area and perimeter for each of the following rectangles.

1. 2. 3.

4. 5. 6.

7. 8. 9.

10. 11. 12.

13. The rectangle below has an area of 210 cm2. Work out the length of this rectangle.

6m 40mm

14. Given that the perimeter of the rectangle below is 44cm, work out the length of the rectangle.

15. Work out the area for each of the following triangles.

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

16. Given that the area of the triangle below is 28 cm2, work out the base length.

20cm 6cm

8cm 5m

17. A triangle has base length 10 cm and area 50 cm2. 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 cm2

(c) 11 cm 44 cm2

(d) 9cm 72cm2

(e) 5cm 15cm

20. For the triangle below Sharon says that the area is equal to 130cm2.

Explain why she is wrong.

21. Work out the shaded area for each of the following:

(a) (b) (c)

22. Work out the shaded area for each of the following:

(a) (b) (c)

(d) (e)

23. Work out the area for each of the following (all measurement are in cm):

(a) (b) (c)

11 cm 3 cm

13 cm 7 cm

4 cm 4 cm

6 cm 10 cm

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 cm2, find its height.

3. The area of the trapezium below is 36cm2.

Find the value of 𝑥.

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 480m3, 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 630cm3.Given that two of the dimensions are 10 and 7cm, what is the

length of the third dimension?

9. A cuboid has volume 240 cm3. Work out a set of possible dimensions for this cuboid.

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 cm2, and volume 210cm3. How long is the prism?

12. Given that the volume of the solid below is 640 cm3, find its height.

13. Which of the solids below has the greatest volume? Show your working.

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.

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.

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

9. Reflect KLMN in the line 𝑦 = −1 10. Reflect HIJ in the line 𝑦 = 3

11. Reflect STU in the line 𝑥 = 3 12. Reflect ABC in the line 𝑦 = 1

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)

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)

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)

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

2. Label the image A

(b) Translation of

1. Label the image B

(c) Translation of

4. Label the image C

(d) Translation of

2. Label the image D

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

a) Translation of

3. Label the image A

b) Translation of

1. Label the image B

c) Translation of

4. Label the image C

d) Translation of

2. Label the image D

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 L1M1N1

3. Enlarge ABC by a scale factor of 4 centre (3, 2). Label the image A1B1C1

4. Enlarge the object below with centre (-1, 1) by a scale factor 3.

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)

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) (c)

(d) (e) (f)

(g) (h) (i) (j)

Work out the required lengths for each of the following, giving your answers to 2 decimal places.

2. Find a 3. Find b

4. Find AC 5. Find EF

6. Find PR 7. Find LM

8cm 8cm

Q 8.3m

8. Find p 9. Find x

10. Find AC

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.

12.1cm

19 cm 23cm

mathematicsbtrcc.pbworks.com/w/file/fetch/112212487/stage 3 geometry...geometry (4) construction of...

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