Maths Year 9 (Part 2)

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Area of a triangle

Area = 1/2 ab sinC

Must have an angle between 2 sides

  • Area = 1/2 x 7.3 x 5.8 x sin37 = 12.7
  • Finding side a = 34.9 (area) = 1/2 x a x 27 x sin15
  • a = 69.8 = a x 27 x sin15
  • a = 69.8 -- (27 x sin15)
  • a = 10
  • Finding angle c: 12 = 1/2 x 4 x 6.2 x sinC
  • 24 = 4 x 6.2 x sinC
  • 24 -- (4 x 6.2) = sinC
  • sinC = 0.967 (Msut be less than 1)
  • C = sin  (0.967) = 75.4
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Constructions

Perpendicular Bisector (Exactly half of the line) : 

  • Draw a straight line
  • Move your compass to the edge & Draw two small arcs
  • Repeat on the other side & connect the points

Perpendicular Line (From anywhere) :

  • Draw a straight line
  • Move your compass to any point on the line & draw an two small arcs
  • Widen the compass to both arcs
  • Steps 3-5 previously

Angle Bisector :

  • Draw any angle
  • Move the compass to the meeting point of the lines & draw an arc on both lines
  • Move the compass to both arcs and draw another arc
  • Draw a line where the second arcs meet
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Constructions (Part 2) & Loci

Triangles:

  • Draw a base line (6cm)
  • Set the compass to 3cm, move it to the edge of the line & draw an arc
  • Repeat step 2 with the compass set at 4cm
  • Connect lines

60  Angle (Equilateral Triangle):

  • Draw a 6cm line
  • Move the compass to the edge, set it to 6cm and draw an arc on both sides
  • Connect lines

Fixed Point:

  • Locus is in a circle
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Loci (Part 2)

2 Fixed Points:

  • Equal from both points
  • Do a perpendicular bisector
  • Locus is a straight line

Equidistant (Same distance) from 2 lines:

  • Angle bisector
  • Locus is a straight line

Locus From a line:

  • Parallel lines above & below
  • Semicircles at the end
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Limits of Accuracy

Any measurement we make is rounded to some degree of accuracy, including:

  • Nearest metre
  • Nearest litre
  • 2 d.p.

There are two limits to every rounded value, an upper and lower (min & max):

  • 1.5 (2 d.p.) - 1.45  ,  1.55
  • 220 (nearest 10) - 215  ,  225
  • Eg. A room is 6.4 x 4.3m
  • The smallest the room could be is 6.35 x 4.25 = 26.9875 (don't round), repeat for largest
  • For division:
  • To calculate the Max amount: upper limit / lower limit
  • Min amount: Lower limit / upper limit
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Factorising Quadratics

Using this method, you can factorise expressions in the form:

ax  + bx + c

  • Eg. x  + 7x + 10
  • First, multiply 2 number to get 10 ( 2,5 or 10,1 or -2,-5 or -10,-1)
  • Second, add 2 numbers to get 7 ( 1,7 or 2,5 or -2, 9 etc)
  • Third, see which two number are in both lists ( 2,5 ) :
  • (x + 2) (x +5)
  • If the number on the end is positive (+10), the numbers have the same sign
  • If the number on the end is negative (eg. -6), the numbers have different signs
  • Eg. x  - a  
  • To factorise, we use   a :
  • (x + a) (x - a)
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Solving Quadratics // Area formulae

Solving Quadratics:

  • x  - 3x - 18 = 0
  • Factorise first: ( x - 6) (x + 3) = 0
  • To make 0, one bracket must equal 0
  • You get two answers (for each bracket):
  • x = 6 or x = -3 (swap signs)

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Area Formulae:

A = b x h

A =     r  

A = 1/2 (a + b) h

A = 1/2 b x h

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Arcs, Sectors & Volume (Formulae)

A. of sector = (    / 360) x    r  :

A. of sector = (60 / 360) x    12  = 75.3 cm

Length of arc = (    / 360) x 2   r:

Length of arc = (45 / 360) x 2   r = 8.64

V. of prism = cross-section of area x depth

V. of cylinder =    r  h

V. of cone = 1/3   r  h

V. of square based pyramid = 1/3 b x h x l

V. of sphere = 4/3   r 

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Cones

Surface area of a cone:

Area of the curved surface:

  • (    / 360) x   r    OR
  •    r x l
  • l  = r  + h
  • h  = l  - r
  • r  = l  - h
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Similarity

  • We know that these triangles are similar because they have the same properties
  • If you time side DF by 3, you get line AC.
  • This means that these triangles have a scale factor of 3.
  • The angles are all the same
  • S.f. = 25 -- 5
  • BC = 4 x 5 = 20
  • FC = 20 - 4 = 16
  • BD = 15 -- 5 = 3
  • S.f. = 30 / 12 = 2.5
  • a = 8 x 2.5 = 20
  • b = 6 x 2.5 = 15
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