Angles Area and perimeter - Abraham Darby Academy · Angles Area and perimeter 2 Algebra 3 4 ......

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Angles Area and perimeter

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Algebra

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Triangles

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Statistics

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Probability

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Solids Graphs

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Bearings

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Angles

Foundation Higher

Check

B

D

C

A

Lines AB and CD are parallel.

EF is perpendicular to AB.

Angle FAE = 47o

Statements

1. Angle AFC is 47o

2. EF is perpendicular to CD

3. Angle AFE is 47o

47o

F

E

Which statements are true?

Statements 1 & 2 are true

Statement 3 is false.

Angle AFE = 43o

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B

D

C

A

Lines AB and CD are parallel.

AF = AE.

Angle AFE = 70o

Statements

1. Angle FAE is 40o

2. Angle EFD = 70o

3. Angle AFC is 40o

70o

F

E


All 3 statements are true.


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Chords AD and CB intersect at the centre of the circle, O.

Angle CDA = 35o B

D

O

C

A

35o

Statements 1. Angle BCD = 35o

2. Angle CBD = 55o

3. Angle AOC = 70o


All three statements are true.

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Chords AD and CB intersect at point E.

Angle CDA = 40o

B

D E

C

A

40o

Statements 1. Angle DAB = 40o

2. Angle ABC = 40o

3. Angle AEC = 80o


Statement 1 is false because the lines AB and CD are not parallel

Statement 2 is true (angles on same arc)

Statement 3 is false because E is not the centre of the circle

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Area & Perimeter

Foundation Higher

This large rectangle is made from 4 smaller rectangles, each measuring 2cm by 4cm

Check

Statements 1. The perimeter of a small rectangle is 12cm 2. The perimeter of the large rectangle is 48cm 3. The area of the large rectangle is 24cm2


Well done if you got it right

Only statement 1 is true. The large rectangle measures 8cm by 4cm so its perimeter is 24cm and its area is 32cm2

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This shape is made from 6 rectangles, each measuring 3cm by 5cm and a red rectangle

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Statements 1. The perimeter of the red rectangle is 18cm 2. The area of the red rectangle is 14cm2 3. Less than 10% of the whole shape is red.


Well done if you got it right

Statements 1 & 2 are true.

Statement 3 is not accurate.

Over 13% is red.

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Statements 1. The area of the trapezium is 15cm2 2. BC is 7 cm 3. The perimeter of the trapezium is 18cm

D

B

C

A ABCD is a trapezium. AB = AD = 3cm DC = 7cm BAD = ADC = 90o

3cm

7cm

3cm




Statement 2 is false. You need to use Pythagoras’ Theorem

to find the length of BC.

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Statements 1. Angle ABC = 70o 2. Perimeter of the rhombus = 40 cm 3. Area of rhombus = 96cm2

D

B

C

A ABCD is a rhombus Diagonal AC = 12cm Diagonal DB = 16cm

Angle BAC = 55o E

55o


All 3 statements are true. The diagonals of a rhombus bisect

each other at right angles so you can use Pythagoras’ Theorem to calculate

the length of a side.

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Algebra

Foundation Higher

Add two touching bricks to obtain the expression in the brick above them.

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7x

13x

6x x

Statements 1. 7x + 6x = 13x 2. 6x + x = 7x 3. The value of x must be 5

?

100


All 3 statements are true

20x = 100

x = 5

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Add two touching bricks to obtain the expression in the brick above them.

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3x + 7

4x + 2

x - 5 5x + 3

Statements 1. (3x + 7) + (x – 5) = 4x + 2 2. (x – 5) + (5x + 3) = 6x + 2 3. The value of x must be 6

?

60


Was it easy?


Statement 2 is not true because (x – 5) + (5x + 3) = 6x - 2

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This is a 12m by 10m rectangular garden. On two sides is a path of x metres width. The remaining two thirds of the garden is grass.

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x m

x m

10 m

12 m Statements 1. x < 10 2. x 2 - 22x + 40 = 0 3. The width of the path is 2 cm




Statement 3 is not true because the width is 2 metres (not cm)

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This is a 12m by 10m rectangular garden. A path of width y metres surrounds a lawn at the centre of the garden.

Check

y m

8 m

12 m

Statements 1. The area of the path is 64 cm2

2. The area of the path is 44y cm2 3. y < 2

y m

The area of the path is twice the area of the lawn.


Only statement 1 is true.

The area of the path is

8y 2 – 80y + 192

so y = 2

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Triangles

Foundation Higher

15 m

ABC is a right angled triangle

Check

B

Statements 1. You can use Pythagoras’

Theorem to find AC 2. AC2 = 152 + 82

3. AC =289 m

8 m

A

C




AC2 = 289

AC = √289 = 17 m

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12 feet

A 13-foot ladder leans against a 12-foot wall so that the top of the ladder just reaches the top of the wall.

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13 feet Statements

1. You can use Pythagoras’ Theorem to find x.

2. x2 = 122 + 132

3. The value of x is 5

A bottom of the ladder is x feet from the base of the wall.

x feet





The correct equation is 132 = 122 + x 2

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13 feet Statements 1. Tan Θ = 12/13 2. Tan Θ = 2.4 3. The value of Θ is 67.3o

to 1 decimal place

A ladder makes an angle of Θ with the ground.

A 13-foot ladder leans against a 12-foot wall so that the top of the ladder just reaches the top of the wall.

12 feet

Θ


Only statement 2 is true.

Statement 1 is false because

Sin Θ = 12/13

Statement 3 is false because

Θ is 67.4o to 1 d.p.

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Statements 1. tan 12o = 75/x 2. 75cm = 7.5 m 3. x = 3.53 to 3 s.f.

A wheel chair ramp rises by 75 cm. The angle of the ramp is 12o. The length of the ramp is x metres.

75cm 12o

x m


Only statement 3 is true. 75 cm = 0.75 m

So tan 12o = 0.75/x

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Statistics

Foundation Higher

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Age in years Frequency 7 2 8 9 10 11

3 4 7 5

This table shows the ages of children in a club.

Statements 1. The mode is 7 years 2. The median is 9 years 3. The mean is 45 ÷ 21


None are true The mode is the most common age = 10 years

The median is the middle age in order (11th out of 21) Median = 10 years.

Sum of the ages = 14 + 24 + 36 + 70 + 55 So the mean is 199 ÷ 21 = 9.48

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Age in years Frequency 0 < x ≤ 20 3 20 < x ≤ 40 40 < x ≤ 60 60 < x ≤ 80 80 < x ≤ 100

9 4 6 3

This table shows the ages of people in a village.

Statements 1. The ages of 25 people are recorded 2. The modal class is 20 < x ≤ 40 3. The best estimate for the median is 50



Statement 3 is not true. The median value is the first of

the 4 values in the 40 < x ≤ 60 interval.

So 45 is a better estimate.

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This table shows the number of books read by a class of pupils during a Charity Readathon

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Number of books Frequency 5 13 6 7 8 9

9 4 3 1

Statements 1. Altogether there were 30 pupils 2. Altogether, they read 35 books 3. Mean = 6 books


Try another


Statement 2 is not true. The pupils read 180 books altogether

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This table shows the ages of people who live in a village.

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Number of books Frequency Age in years Frequency 0 < x ≤ 20 3 20 < x ≤ 40 40 < x ≤ 60 60 < x ≤ 70 70 < x ≤ 100

9 4 6 3

Statements 1. A good estimate for the mean is 45 years 2. The most densely populated class is 20 < x ≤ 40 3. On a histogram the relative frequency for the

interval 60 < x ≤ 70 would be 0.6



Statement 2 is not true. 20 < x ≤ 40 is the most populated

but not the most densely populated

(relative frequency is 0.45)

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Probability

Foundation Higher

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Statements 1. Probability that the lights are red or amber = 0.36 2. Probability that they are green = 0.55 3. Probability that they are not red = 0.75

The probability the lights are

red is 0.35

Each day the school bus passes a set of traffic lights.


The probability the lights are amber is 0.1

Statements 1 is not true

( 0.35 + 0.1 = 0.45)

Statement 2 is true

prob(green) = 1 – 0.45

Statement 3 is not true.

prob (not red) = 1 – prob (red) = 0.65

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Statements 1.The probability of picking a blue ball from bag A is 0.3 2.You are more likely to pick a blue from bag B than bag A 3.You are more likely to pick green from bag A than bag B

Bag A Bag B

A ball is picked at random from each bag


Statements 1 is true.

Statement 2 is not true (probabilities are 0.3 and 0.25)

Statement 3 is not true. (probabilities are 0.6 and 0.625)

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The probability that the

first lights are red is

0.3

Statements 1. The first set of lights are more likely to be green

than the second set. 2. The probability that both lights are red is 0.6 3. The probability that only one set is red is 0.38

The probability that the

second lights are red is

0.2

Each day the school bus passes two sets of traffic lights.


Statements 1 & 2 are not true.

Prob (RR) = 0.2 x 0.3 = 0.06

Statement 3 is true.

prob (one red) = prob (GR) + prob (RG)

= (0.7 x 0.2) + (0.3 x 0.8)

= 0.14 + 0.24 = 0.38

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Statements 1. The probability of picking at least one yellow = 0.65 2. The probability that both are yellow = 0.1 3. If the ball from bag A is put into bag B before a ball

ball is taken from bag B then the probability that the ball from bag B is yellow = 0.28

Bag A Bag B

Two balls are picked at random – one ball from each bag


Statement 1 is not true.

Prob (one yellow) = p(yy’) + p(y’y)

= (0.4 x 0.75) + (0.6 x 0.25) = 0.45


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Solids

Foundation Higher

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This box has a 4 cm by 5 cm rectangular base and a height of 3 cm

4cm

Statements 1. The area of the top is 20 cm2

2. The area of the front face is 15 cm2

3. The total surface area is 47 cm2


5cm

3cm


but statement 3 is not true because

total surface area is 94cm2

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Each of these boxes has a 4cm by 5 cm rectangular base The green box has a height of 3cm The blue box has a height of h cm

hcm 3cm

4cm 4cm 5cm 5cm

Statements 1. Volume of the green box is 60 cm3

2. Volume of the blue box is 20h cm3

3. If the volume of the blue box is 76 cm3 then its height is 3.8 cm




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8cm

Check

C

A B

D

E

9cm

12cm

This is a right-angled triangular prism

Statements 1. Area of cross section is 108 cm2

2. Volume = 864 cm3

3. Total surface area = 396 cm2

F


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Only statement 3 is true

Statements 1& 2 is not true. The area of the cross-section is

½ x 9 x 12 = 54 cm2

and so the Volume is 432 cm3

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12 m

Check

C

A B

D

E

8cm

9cm

This is a right-angled triangular prism

Statements 1. Journey A → F → E → D = 29cm 2. Journey A → F → D = 23cm

3. Journey A → B → E = 29cm

F


Journeys between two points are made along the shortest paths

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They are all true.

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Graphs

Foundation Higher

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y

x

A

B

Statements 1. B = (-2, -1) 2. The mid point of line AB = (1, 1) 3. AB = √52 units

A = (4, 3)


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y

x

R

T

Statements 1. PQ is parallel to TS 2. PQST is a rhombus 3. Angle RTS is a right angle

P = (1, 3)


S

P

Q


Statement 2 is not true. The shape is a parallelogram

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y

x

A

B Statements 1. Gradient of AB is 1 2. Equation of line AB is y = x + 1 3. Length AB = 5√10 units

A = (-2, 3)


None of the statements are true.

The gradient of the line is -1

Hence, the equation of the line is

y = -x + 1 or x + y = 1

AB = √50 units = 5√2 units

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y

x

Statements 1. A = (-2,0) 2. Equation of line BC is y = 8 3. Equation of line through C perpendicular

to AB is y = 14 - 2x

The equation of line AB is

y = ½x + 2


A

B

C

Statements 1 & 2 are not true

A = (-4, 0)

The equation of BC is x = 8

Statement 3 is true.

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Bearings

Foundation Higher

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North North

Island lighthouse

Port

A positions of a Port, a lighthouse and an island are shown on this map. The Island is on a bearing of 045o from the Port.

45o

Statements 1. The Island is East of the lighthouse. 2. The Island is North-East of the Port

3. The Port is South-East of the Island


Only statements 1 & 2 are true.

Statement 3 is false because the Port is South-West of the Island

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North North

Island lighthouse

Port

16 km 34 km

A boat leaves the Port and sails 16km North, goes round a lighthouse and sails East to the Island. The island is 34km from the Port on a bearing of 062o

62o

Statements 1. The Island is 30km from the lighthouse 2. The bearing of the Island from the lighthouse is 090o

3. The bearing of the Port from the Island is 242o



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North North Harbour

Buoy

Port

30 km

A boat leaves the Port at Noon, sails 30km North, goes round a Buoy and sails to the Harbour on a bearing of 080o.

The Harbour is on a bearing of 060o from the Port.

The average speed of the boat is 20 km/hr

60o

Statements 1.The bearing of the Buoy from the Harbour is 260o

2.The Harbour is 86.4 km from the Port (3sf) 3.The Boat arrives at the Harbour at 17:18

80o



You need to use the Sine Rule

for this question.

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North North

Wreck

Reef

Port

20 km

The Reef is 20km North of the Port. The wreck is 30 km from the reef on a bearing of 120o

Statements 1. The Wreck is 26.5km from the Port (to 3sf) 2. The bearing of the Wreck from the Port is 079o to the

nearest degree 3. The area of sea within the triangle is 260 km2 (3sf)

120o


30 km


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Angles Area and perimeter - Abraham Darby Academy · Angles Area and perimeter 2 Algebra 3 4 ......

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Transcript of Angles Area and perimeter - Abraham Darby Academy · Angles Area and perimeter 2 Algebra 3 4 ......