BEG CSE Shape and Space revision

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BEG CSE Shape and Space revision Powered By Docstoc
					Intermediate Tier – Shape and space revision

         Contents :   Angle calculations
                      Angles and polygons
                      Bearings
                      Units
                      Perimeter
                      Area formulae
                      Area strategy
                      Volume
                      Nets and surface area
                      Spotting P, A & V formulae
                      Transformations
                      Constructions
                      Loci
                      Pythagoras Theorem
                      Similarity
                      Trigonometry
                      Circle angle theorems
                                      “F” angles are equal             Use the
Angle calculations                               570                   rules to
Angles in a half turn = 1800                                          work out
                                                                      all angles
                                                    h   i

      720 a       210               Angles in a triangle = 1800

Angles in a full turn =   3600               120            j
                                                                350
              b
     1350
            1620                     Angles in a quadrilateral = 3600
                                             k                  730
Opposite angles are equal
                                              980               l
         1530
        e d c
                                 Angles in an isosceles triangle

“Z” angles are equal                     m
                                                            80
         f        g
        420
                                     There are 3 types of angles in regular
 Angles and polygons                               polygons

Angles at =       360           Exterior =   360            Interior = 180 - e
the centre       No. of sides   angles     No. of sides     angles


             c
        c c
                                                   e
        ccc                                                e      i

 Calculate the value of c, e and i in
 regular polygons with 8, 9, 10 and
 12 sides
 Answers:
 8 sides = 450, 450, 1350
                               To calculate the total interior
 9 sides = 400, 400, 1400     angles of an irregular polygon      Total i
 10 sides = 360, 360, 1440   divide it up into triangles from 1   = 5 x 180
 12 sides = 300, 300, 1500      corner. Then no. of x 180         = 9000
 Bearings          A bearing should always            A bearing is an
                   have 3 figures.                    angle measured in
                                                      a clockwise
What are these bearings ?                             direction from due
                                                      North

                                                                 N
                                 Here are the steps to get
                                 your answer                          Bristol
                                                                       2360


                                   N
                                       560       Notice that there is a 1800
                                                 difference between the
                                                 outward journey and the
                          Bath                   return journey
                                                      0560
What is the bearing of Bristol from Bath ?                           2360
What is the bearing of Bath from Bristol ?
                                 x 1000        x 100         x 10
Units

 Learn these metric
    conversions               kl
                             kg
                             km            m
                                           l            cl
                                                       cg
                                                       cm           ml
                                                                    mg
                                                                    mm
                       Length
                       Weight
                      Capacity
                                  ÷ 1000       ÷ 100         ÷ 10



                                    Imperial  Metric
        Learn these rough             5 miles  8 km
        imperial to metric            1 yard  0.9 m
           conversions              12 inches  30 cm
                                     1 inch  2.5 cm
Perimeter
                                                           The
                           Circumference =  x D
Perimeter = 4 x L
of a square                                           perimeter of
                           of a circle
                  Be prepared to leave answers a shape is
                                                      the distance
                 to circle questions in terms of around its
                 26m
              especially in the non-calculator exam      outside
        6.5m                             5m
                                                       measured
                                                        in cm, m,
Perimeter      = 2(L + W)        Perim = D + ( x D)  2 etc.
                                              31.4m
of a rectangle
                                 Perim = 15 + ( x 15)  2
                                 Perim = 15 =  x D
         2m                    Circumference + 7.5
                     15cm
                            of a semi-circle    2
                                                7.85m
              7.2m
                                                    Perimeter = ?
                                     4.71m
               18.4m            1m    3m       1m   = 7.85 + 4.71 + 1 + 1
                                                    = 14.56m
                        The area of a 2D shape is the amount of space
Area formulae              covered by it measured in cm2, m2 etc.
Area of = L x W
square              Area of       =bxh          Area of   = (a + b) x h
                  Beparallelogramto leave
                     prepared                   Trapezium
                                               answers           2
                                                         6m
    49m2
               to circle questions in terms of  4m
                          4m
                              40m2
                                    5m

     7m      especially in the non-calculator exam
                              10m
                                                                5m
                                                                 2
Area of = L x W                                            2m 16m
                          Area of = b x h
rectangle
                          triangle   2
                                  Area = ( x r x r)  2of =  x r2
                                                  Area
                  2m
    18m2
                                  Area = ( x 5 x circle2
                                   9m
                                                   5) 
     9m                10cm     8mArea = 12.5
Area of = b x h                  24m2
rhombus                         6m
                                                           8m
         42m2                      7m                     50.24m2
    6m                   3m
                               7.5m2
   7m                                  5m
Area strategy    What would you do to get the area of each of
                   these shapes? Do them step by step!
1.   1.5m
                                                       4m
                           2.                    3.
                                2m
9m                                        10m
                            7m
                 2m

            8m                                         6m

        4.                                      3m
                                     5.


                      6m                                1.5m



                                                6m
Volume       The volume of a 3D solid shape is the amount of
                 space inside it measured in cm3, m3 etc.

                                 Volume of = Area at end x L
Volume = L x L x L
                                 a prism
of cube

                                                         A = 14m2
                3m
                                 4m
   27m3                               56m3




                                   Volume of = ( x r2) x L
Volume of = L x W x H
                                   cylinder
cuboid
                     2m
      42m3           3m          7m                 384.65m3
       7m

                                                   10m
                                                      6
 Nets and surface area
                                                    12cm2
                                                    12cm2         4cm2
                                          2
                                     2   4cm2       12cm2
   Cuboid 2 by 2 by 6
                                                                  Net of the cuboid
 Volume = 2 x 2 x 6 = 24cm3                         12cm2

    To find the surface area of a               Total surface area
   cuboid it helps to draw the net              = 12 + 12 + 12 + 12 + 4 + 4
                                                = 56cm2

Find the volume and surface area of these cuboids:
                                                             3.
 1.                       2.



        5 by 4 by 3               6 by 6 by 1                      5 by 5 by 5
  V = 5 x 4 x 3 = 60cm3        V = 6 x 6 x 1 = 60cm3        V = 5 x 5 x 5 = 125cm3
  SA = 94cm2                   SA = 96cm2                    SA = 150cm2
 Spotting P, A & V formulae
                                  r(+ 3)                4rl
                                          P                       A
                                                  r(r + l)
Which of the following
expressions could be for:
                                                              A
(a) Perimeter
(b) Area
                                   1d2
(c) Volume        4r2             4      A            4r3
                 3     A                               3
                                r + ½r                       V
  4l2h                                    P
        V         1rh                                   1r2h
                  3 A               r + 4l              3
   1r
                                                                  V
                                                   P
   3 P           4r2h      V       3lh2      V
                                                        rl A
Transformations
1. Reflection     y

 Reflect the
 triangle using
 the line:
     y=x
 then the line:

                      x
    y=-x
 then the line:
     x=1
Transformations      Describe the rotation of A to B and C to D

2. Rotation                            y

When describing
a rotation always
state these 3
things:


                           B
• No. of degrees                             C
• Direction
• Centre of
rotation
                                                                  x
e.g. a rotation of
900 anti-
clockwise using
a centre of (0, 1)              A
                                                   D
                                    What happens when we translate a shape ?
Transformations                   The shape remains the same size and shape and
                                       the same way up – it just……. slides
                                                                         .
3. Translation


                         Horizontal translation            Use a vector
                                                            to describe
                                                                            3
                                                           a translation   -4
  Vertical translation




                           Give the vector for
                             the translation
                               from……..
                                                                     D
                                                  6
                           1.    A to B           0         C
                                           6
                           2.    A to D
                                           5
                           3.    B to C           -3
                                                  4    A             B
                           4.    D to C -3
                                        -1
                          Enlarge this shape by a scale
Transformations           factor of 2 using centre O
4. Enlargement        y




                                                     x
                  O
Constructions
                            Perpendicular
    Have a look at these    bisector of a line
   constructions and work
     out what has been
            done


                             Triangle with 3
                             side lengths
                900
                                 Bisector of
                                 an angle


            600
     Loci     A locus is a drawing of all the points which satisfy a rule or
              a set of constraints. Loci is just the plural of locus.

A goat is tethered to a peg in the
ground at point A using a rope 1.5m long

1.     Draw the locus to show                                A
       all that grass he can eat
                                                         1.5m



A goat is tethered to a rail AB using a rope
(with a loop on) 1.5m long                                1.5m
2.     Draw the locus to show all that               A                    B
       grass he can eat
                                                                        1.5m
                  Shapes are congruent if they
  Similarity      are exactly the same shape
                  and exactly the same size

  Shapes are similar if they                                   How can I
  are exactly the same shape                                  spot similar
  but different sizes                                         triangles ?

                                           These two triangles are similar
                                           because of the parallel lines


       Triangle C



                    Triangle B
    Triangle A

All of these “internal” triangles are similar
to the big triangle because of the parallel lines
Similarity                                            Triangle 2
These two triangles are similar.Calculate length y



     y = 17.85  2.1 = 8.5m
                    Same multiplier
                           x 2.1           17.85m              15.12m



 y           7.2m




Triangle 1                                    x 2.1
                              Multiplier = 15.12  7.2 = 2.1
                        Calculating the Hypotenuse
Pythagoras Theorem
                            D                        Hyp2 = a2 + b2
     How to spot a                                   DE2 = 212 + 452
           Be prepared to leave your?




                            21cm
                                        answer       DE2 = 441 + 2025
      Pythagoras
           in surd form (most likely in the
       question                                      DE2 = 2466
            non-calculator exam)                 DE = 2466
  Right angled              F     Hyp2 = a2 + bE DE = 49.659
                                  45cm
            D
                                               2

     triangle                        the
                           Calculate 2 size 2    DE = 49.7cm
                                  DE = 32 + 6
                        ?    of DE to 1 2d.p.
            3cm



                                    DE = 9 + 36
  No angles                         DE2 = 45
                        Calculating a shorter side     Hyp2 = a2 + b2
   involved                         DE = 45              162 = AC2 + 112
            F
  in question       6cm         E DE = 9 x 5
                                    A                   256 = AC2 + 121
            Calculate the size
                                        ?
                                    DE = 35 cm
            of DE in surd form                     256 - 121 = AC2
  How to spot the                                      135 = AC2
    Hypotenuse
                                                        135 = AC
                         B         16m         C 11.618 = AC
 Longest side &           Calculate the size
    opposite              of AC to 1 d.p.               AC = 11.6m
Pythagoras Questions

Look out for the following Pythagoras questions in disguise:

     y         Find the distance                  Finding lengths
                                                  in isosceles
             x between 2 co-ords
                                                  triangles


 x             x


                   Finding lengths                Finding lengths
                   inside a circle 1              inside a circle 2
                   (angle in a semi               (radius x 2 =
                   -circle = 900)                    isosc triangle)
         O                             O
                            Calculating an angle
 Trigonometry
                                 D                              SOHCAHTOA
        How to spot a                                           Tan  = O/A




                                 26cm
                                             H                  Tan  = 26/53
        Trigonometry
           question                                             Tan  = 0.491
                                 O               
                                                                     =   0

    Right angled                   F    A 53cm             E
       triangle                  Calculate the size
                                 of  to 1 d.p.

     An angle               Calculating a side
     involved
    in question                          D                     SOHCAHTOA
                                                               Sin  = O/H
                                 O
                                             ?   A
                                                               Sin 73 = 11/H
•Label sides H, O, A
•Write SOHCAHTOA                                 730               H = 11/Sin 73
•Write out correct rule
•Substitute values in
                             B          H              C          H =         m
                              Calculate the size
•If calculating angle use
                              of BC to 1 d.p.
  2nd func. key
Circle angle theorems       Rule 1 - Any angle in a
                                     semi-circle is 900
                        A
F
                                       Which angles
                                       are equal
                                       to 900 ?
              c
                                B

                            C
    E
                  D
Circle angle theorems

Rule 2 -     Angles in the same segment
             are equal




    Which angles are
      equal here?




                                          Big fish ?*!
Circle angle theorems            Rule 3 - The angle at the centre
                                          is twice the angle at the
                                          circumference



                                                          c
         c                   c




  An arrowhead          A little fish            A mini quadrilateral




             c                                   Look out for the
                                      c          angle at the centre
                                                 being part of a
                                                 isosceles triangle

     Three radii
Circle angle theorems

 Rule 4 - Opposite angles in a cyclic
          quadrilateral add up to 1800



                 D
                        C          A + C = 1800
        A
                                         and
                        B
                                   B + D = 1800
Circle angle theorems

Rule 5 -     The angle between the tangent
             and the radius is 900




                        c



                            A tangent is a line which
                            rests on the outside of the
                            circle and touches it at one
                            point only
Circle angle theorems

Rule 7 -     Tangents from an external point
             are equal (this usually creates a
             kite with two 900 angles in…..
               …… or two isosceles triangles)


                        900



                   c


                        900

				
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