# 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)

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**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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