YEAR 11 REVISION BLASTER UNIT 2 STAGE 2. GCSE MATHS REVISION UNIT 2 (stage 2) – all about: SHAPE & ALGEBRA & NUMBER Circle theorems Angles & lines & shapes.

YEAR 11

REVISION BLASTER

UNIT 2 STAGE 2

GCSE MATHS REVISION

UNIT 2 (stage 2) – all about:SHAPE & ALGEBRA & NUMBER• Circle theorems• Angles & lines & shapes• Simplifying & factorising• Linear graphs (y=mx±c)• Percentages

CIRCLE THEOREMS

The angle made at the centre of a circle (by an arc or chord) is double the angle made at the circumference by the same arc or chord.

2xo

2xo 2xo 2xo

2xo 2xo 2xo 2xo

xo

xo

xoxo

xo xo xo xo

o oo o

o o o o

Angle at centre is a reflex angle

Another Theorem

The Angle at Centre (Arrow) TheoremTheorem 1

o Diameter

90o angle in a semi-circle90o angle in a semi-circle20o angle sum triangle

90o angle in a semi-circle

e

o

a

b

c

70o

d

30o

Find the unknown angles below stating a reason.

angle a = angle b = angle c = angle d = angle e =

60o angle sum triangle

Angle in Semi-Circle TheoremTheorem 2

This is a special case of Theorem 1: angle in a semi-circle is always 90°

Th2

Angles subtended by an arc or chord in the same segment are equal.

xo xo

xo

xo

xo

yo

yo

Th3

The Chord (Bow) TheoremTheorem 3

The angle between a tangent and a radius is 90o. (Tan/rad)

o

The Tangent TheoremTheorem 4

The Alternate Segment TheoremTheorem 5

The angle between a tangent and a chord through the point of contact is equal to the angle made by that chord in the alternate segment.

xo

xo

yo

yo

45o (Alt Seg)

60o (Alt Seg)

75o angle sum triangle

45o

xo

yo

60o

zo

Find the missing angles below giving reasons in each case.

angle x = angle y = angle z =

Th5

Cyclic Quadrilateral TheoremTheorem 6

The opposite angles of a cyclic quadrilateral are supplementary. (They sum to 180o)

w

y

x

z

Angles x + z = 180o

Angles y + w = 180o

r

p

s

q

Angles p + r = 180o

Angles q + s = 180o

All corners touch the circle circumference

The Two Tangent TheoremTheorem 7

From any point outside a circle only two tangents can be drawn and they are equal in length.

P

T

UQ

R

PT = PQ

P

T

U

Q

R

PT = PQ

Th7

Circle Theorem Answers

1. (a) 58°(b) 2 angles are the same (=58°)

2. (a) (i) 70°(opposite angles of) cyclic quadrilateral

(ii) 140°angle at centre is twice angle at circumference

3. (a) (i) 40°(ii) 140°

(b) Right angle at A / C (Tangent Theorem): at B 2 x 50° angles

4. x = 52°; y = 76° and z = 58°

Angles

Which angles are equal to each other?

ab

cd

ef

gh

Alternate angles are

equal

Alternate angles are

equal

a

b

a = ba = b

Look for an F-shape

Look for a Z-shape

Corresponding angles are

equal

Corresponding angles are

equal

a

b

a = ba = b

Look for a C- or U-shape

Interior angles add up to 180°

Interior angles add up to 180°

a

b

a + b = 180°a + b = 180°

ab

c

Any exterior angle in a triangle is equal to the sum of the two opposite interior

angles.

Any exterior angle in a triangle is equal to the sum of the two opposite interior

angles.

a = b + c

We can prove this by constructing a line parallel to this side.These alternate angles are equal.These corresponding angles are equal.

bc

Calculate the size of angle a.

a28º

45º

Hint: Add another parallel

line.

Hint: Add another parallel

line.

a = 28º + 45º = 73º

Angles & Lines & Shapes Answers

1. (a) x = 60.5(b) Not parallel because angle A is not equal to x,

or because angle B is not equal to x,or because alternate (or Z) angles are not equal,or because corresponding angles are not equal.

2. (a) (i) x = 74

(ii) y = 36°(b) No. Because triangle ABD is not isosceles.

or Because y is not equal to 38°.3. (a) 30°

(b) 40°

4. a = 50° and b = 110°

AD BREAKYear 11 Unit 2 Exam(s). Monday 15th November-

NEXT WEEK!!

Don’t Forget: CALCULATORS

PEN

PENCIL

RULER

REVISE

Algebra

Expanding and simplifying

• (x+2)(x+3)

x² +3x

+2x +6

= x²

= x² + 5x + 6

+ 3x+ 2x +6

Factorising

• x² + 7x + 12

+12 x

x

+6

+2

+6x

+2x

= x² + 8x +12

= x² + 6x +2x +12

+4

+3

+4x

+3x

= x² + 4x +3x +12

= x² + 7x +12

So,

x² + 7x + 12 = ( )( )

Which simplifies to: 20a³b³

• 4ab x 5a²b²Simplifying

10ab²

First, focus on the 4a³b x 5a²b²

Rearranging gives: 4x5 x axa² x bxb²

20a³b³

10ab²

= 20 a³ b³

10 a b²= 2a²b

Simplifying & Factorising Answers

1. (a) 3x(x – 2y)

(b) (y – 7)(y – 2)

2. (a) (x – 7)(x – 3)

3. (a) 24abc

4. a4 c–1 or

5. (b) 5xy(x + 3y2)

6. (b) 2a2 b2 c

c

a4

STRAIGHT LINE GRAPHS

General equation of straight line.

y = mx + c

y = mx + c

Gradient/ Steepness of the line.

Y-intercept. (where the graph crosses the y axis)

Plotting a straight line…y = 2x -1Pick 3 values of x. (e.g -2, 0, 2)

SUBSTITUTE these into the equation (y=2x-1)

xy

when x = -2,

Y = (2x-2) – 1

Y = - 4 – 1

Y = -5

When x = 0,

Y = (2x0) – 1

Y = 0 – 1

Y = -1

When x = 2

Y = (2x2) – 1

Y = 4 – 1

Y = 3

-2 0 2

-5 -1 3

Find the gradient:

Draw a triangle under the graph.

Make sure the height and width of the triangle are WHOLE numbers

Rise = 8

Run = 4

Gradient = Rise = 8

Run = 4

= 2

Plot these pairs of co-ordinates. Join up the points, and extend across the axes.

y =

Gradient/ Steepness of the line.

Y-intercept. (where the graph crosses the y axis)

mx+ c

2 -1

Key Facts•Parallel lines have the SAME

gradient

• Graphs with a negative gradient slope DOWNHILL

•Graphs with a positive gradient slope UPHILL

•The bigger the gradient the steeper the slope

Linear Graphs Answers

1. 0.8, 4/5, or equivalent2. (a) y = 0.4x + 18

(b) 583. (a) (– 6, 0)

(b) ½(c) Line drawn parallel to 2y = x + 6

4. (i) (–½, 0)(ii) reasonably parallel line crossing y axis below origin(iii) parallel lines do not meet

Percentage Increase

and Decrease

Increase £20 by 15%

Method 1Find 15%10% = £2 5% = £1 so 15% = £3

Now add it on to £20£20 + £3 = £23

Increase by 15% so

1.15 is the multiplier

Method 2Method 2

I need to find 115% I need to find 115% OF original amountOF original amount115% of £20 is 115% of £20 is 1.15 x £20 = £231.15 x £20 = £23

Decrease £60 by 5%

Method 1

Find 5% of £60

10% = £6 5% = £3

Subtract from £60

£60 - £3 = £57

Method 2

A decrease of 5% is same as 95% of original amount

So 0.95 x £60 = £57 95% of original amount means 0.95 is the multiplier

Compound Interest

• £1,000 in bank earns 5% interest per year. How much will you have in 3 years?

After 1 year 1000 x 1.05 = 1050

After 2 years 1050 x 1.05= 1102.50

After 3 years 1102.50 x 1.05 = 1157.63

OR 1000 x 1.05 x 1.05 x1.05=

1000 x 1.05³=£1157.63

1.05 is the multiplier!

Percentages Answers

1.£2140 (allow £140)

2.(a) 1.029

(b) 1 087 401 937 (allow 1 087 000 000)

3.1st year is £20; 2nd year is £20.80

Interest is £40.80 so Amir is wrong

4.(a) 2.04

(b) 6 (windmills)