Transcript Slide 1

Mathematic
(HSC)
Stage 6 - Year 12
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1
Types of Angles
2. Right Angle
1. Acute angles
3. Obtuse Angle
(0o < θ < 90o
4. Straight Angle
(θ = 180o)
θ = 90o
(90o < θ < 180o)
6. Angle of
Revolution
5. Reflex Angle
(180o < θ < 360o)
(θ = 360o)
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Pairs of Angles
1. Vertically Opposite Angle
are equal.
2. Complementary Angles
ao
2. Supplementary Angles
ao
bo
bo
add to 90o.
add to 180o.
3
Angles between Parallel Lines
1. Alternate Angles
Transversal
3. Co-Interior angles
2. Corresponding Angles
Makes a
Z shape.
and
are equal.
Makes a
Makes a
F shape.
and
are equal.
C shape.
and
Add to 180o
4
Types of Triangles
Based on Sides
1. Equilateral triangle.
•All sides equal
•All angles equal (60O)
2. Isosceles triangle.
•Two sides equal
•Two base-angles equal
3. Scalene triangle.
•No sides equal
•No angles equal
Based on Angles
1. Acute angled triangle.
•All angles acute
2. Right angled triangle.
•One angle 90o
3. Obtuse angled triangle.
•One
Obtuse
angle.
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Angle Sums
1. Angle Sum of a Triangle
2. Angle Sum of a Quadrilateral
bo
ao
bo
co
ao + bo + co = 180o
co
ao
do
ao + bo + co + do = 360o
3. Exterior Angle of a Triangle.
4. Angles at a point.
co
bo
ao
ao = bo + co
o
ao c
bo
ao + bo + co = 360o
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Congruence
1. Side, Side, Side.
SSS
2. Side, Angle, Side.
SAS
3. Angle, Angle, Side. 4. Right angle, Hypotenuse, Side.
AAS
RHS
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Similar Triangles
1. Corresponding
angles are all equal.
β
α
γ
β
2. Corresponding sides
are in the same ratio.
α
a
γ
3. Two pairs of sides are in
proportion and their included angles
are equal.
p
θ
q
b
r
φ
s
c
a
x
x
z
=
p
r
y
b
= c
y
z
= q
s
  =  
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Ratio of Intercepts
A
D
B
E
C
F
AB : BC = DE : EF
AB = DE
BC EF
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Pythagoras Theorem
c
a
b
c2 = a2 + b2
You need to be able to:
1. Find the length of the hypotenuse.
2. Find the length of the shorter side.
3. Prove you have a right angle.
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Types of Quadrilaterals
1. Rectangle
4.Parallelogram
2. Square
5. Trapezium
3. Rhombus
6. Kite
You must know their properties
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Types of Regular Polygons
1. Triangle
2. Square
4. Hexagon
3. Pentagon
5. Octagon
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Regular Polygons
a
1. Angle Sum of a Polygon.
= (n – 2) x 180
(n is the number of angles)
2. Interior angle.
b
c
f
e
d
Divide the angle sum by the number of angles.
3. Exterior angle
The exterior angles
of add to 360o.
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Area Formulae
1.Square
A = s2
3.Triangle
A=½bh
s
2. Rectangle
A = LB
L
B
4. Parallelogram
A=bh 6.Trapezium
h
b h
A=½(a+b)h
b
a
5.Rhombus/Kite
7.Circle
2
A=½xy
A=πr
x y
x
y
r
h
b
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Surface Area Formulae
1. Rectangular Prism 2. Cube
3. Sphere
h
b
l
SA = 2(bh + hl + lb)
4. Cylinder
s
SA = 6s2
SA = 4 π r2
5. Cone
r
h
l
h
r
SA = 2 π r (r + h)
SA = π r (r + l)
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Volume Formulae
1. Rectangular Prism 2. Cube
3. Sphere
h
b
l
V = lbh
4. Cylinder
r
s
V = s3
V = Ah
5. Cone
h
V = π r2 h
V = 4 π r3
3
V = 1 π r2 h
3
h
r
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2006 HSC Question 6
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2006 HSC Question 6
(i)
Prove that BAC = BCA
BAC = CAD
[Given]
BCA = CAD
[Alternate angles between parallel lines.]
1
 BAC = BCA
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2006 HSC Question 6
(ii)
Prove that ∆ABP ≡ ∆CBP
1
PBA = PBC
[Given]
BAC = BCA
[See part (i)]
BP
 ∆ABP ≡ ∆CBP
[Common]
[AAS]
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2006 HSC Question 6
(iii)
Prove that ABCD is a rhombus.
APB = BPC
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[corresponding angles in congruent triangles – part ii]
APB + BPC = 180o [straight angle]
 2 x BPC = 180o
BPC = 90o = APB
Diagonals bisect at 90o
[ Square or Rhombus ????]
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2005 HSC Question 5
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2004 HSC Question 2
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2004 HSC Question 6
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