Maths Year 9 (Part 2)
- Created by: carmen.lau
- Created on: 10-06-15 18:30
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
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
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
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
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
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)
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)
////
Area Formulae:
A = b x h
A = r
A = 1/2 (a + b) h
A = 1/2 b x h
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
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
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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