perak state additional mathematics project workjpnperak.moe.gov.my/jpn/attachments/article/2728/item...

PERAK CEMERLANG

PERAK STATE

ADDITIONAL MATHEMATICS

PROJECT WORK

JABATAN PENDIDIKAN NEGERI PERAK

Disediakan oleh :

Sektor Pengurusan Akademik

Jabatan Pendidikan Negeri Perak

2015

Perak State Additional Mathematics Project Work 2015

JPN PERAK Page 1

Worksheet 1 : Getting creative…..

Designing is fun!

It is nice to explore and wise to study geometry. The following quotation says it all beautifully.

“Geometry enlightens the intellect and sets one’s mind right. All of its proofs are very clear and

orderly. It is hardly possible for errors to enter into geometrical reasoning because it is well arranged

and orderly. Thus, the mind that constantly applies itself to geometry is not likely to fall into error. In

this convenient way, the person who knows geometry acquires intelligence.”

Ibn Khaldun [1332 – 1406]

After reading this inspirational quotation, you are now ready to begin this project work by expressing

creatively the beauty of geometry related to triangles.

On a piece of A4 paper, design a birthday gift wrapper. Your design should be based on various types

of triangles: equilateral, isosceles, scalene and right-angled triangles.

Use colour to enhance the beauty of your design.


JPN PERAK Page 2

Worksheet 2 : Getting curious about triangles…..

Case 1: SSS

Diagram 1 shows a triangle ABC with three given sides. [ a, b and c ] This is Case1: SSS.

Make a suitable conjecture about the lengths of the three sides of the triangle.

Test your conjecture by carrying out Exploration 1.

EXPLORATION 1

Materials required

Drinking straws of lengths 5 cm, 6 cm, 7 cm, 11 cm, 12 cm and 15 cm.

[Any other suitable materials like cardboard / plastic strips and satay sticks can be used.]

Procedure

1. Record systematically, in the form of a table, all the possible values of a, b and c that can be

chosen from the six lengths of drinking straws that you have prepared.

You are encouraged to create this table by using ICT. Label it as Table 1.

2. For each set of values of a, b and c, try to use drinking straws of lengths a, b and c to form a

triangle. Can you always form a triangle?

Record your findings in Table1.

Conclusion

1. Does your findings support you conjecture?

Explain your reasoning.

2. State clearly the conclusion that can be drawn from your exploration on the relationship

between the lengths of the three sides of a triangle.

Prove the relationship.

B C

A

b c

a

Diagram 1


JPN PERAK Page 3


Case 2: SAS

Diagram 2 shows a triangle ABC with two sides and one included angle given. [ a , b and C ]

This is Case 2: SAS.

Can a distinct triangle be constructed with these measurements?

Explore Case 2: SAS by carrying out Exploration 2.

EXPLORATION 2

Materials required

A protractor, a pair of compasses and a ruler.

Procedure

1. Prepare Table 2 by choosing at random the values of a, b and C. Include all types of angles

in your exploration.

Table 2

2. For each set of values of a, b and C, construct the corresponding triangle.

In each case, how many different triangles can you construct with the chosen measurements?

Record your findings in Table 2.

Conclusion

State clearly the conclusion that can be drawn from your exploration.

a b C Number of distinct triangles

that can be constructed

B C

A

b

a

Diagram 2


JPN PERAK Page 4


Case 3: SSA

Diagram 3(a) shows a triangle ABC with two sides and one non-included angle given. [b, c and C]

This is Case 3: SSA.

Can a distinct triangle be formed with these measurements?

Explore Case 3: SSA by carrying out Exploration 3.

EXPLORATION 3

Materials required

A ruler, a protractor and drinking straws of lengths 4 cm, 5 cm, 6 cm, 7 cm, 11 cm, 12 cm and 15 cm.

Procedure

1. On a piece of A4 paper, construct Diagram 3(b).

2. By using one drinking straw at a time, try to form a triangle on Diagram 3(b).

Can you always form a triangle?

Record your findings systematically in Table 3.

Table 3

Conclusion

Based on your findings, make a conclusion about how the measures of two sides and one non-

included angle can be used to determine the number of triangles possible.

b c Number of triangles that can be formed

12

B C

A

b c

Diagram 3(a)

30o

Diagram 3(b)

b = 12 cm

C

A


JPN PERAK Page 5

Worksheet 5 : Getting more interesting to solve triangles…..

Case 1: SSS and Case 2: SAS

1. If three sides [Case 1: SSS] or two sides and one included angle [Case 2: SAS] of a triangle

are given, then the remaining side and angles are usually calculated by using a certain

formula.

State and prove the formula.

2. Show how you can use that formula only to solve the following two triangles.

(a) (b)

3. The diagram below shows a quadrilateral ABCD and an arc BFE of a circle with centre A. The

length of arc BFE is 8π cm.

Calculate the length, in cm, of diagonal AC.

C B

A

60o

12 cm

9 cm

R Q

P

5 cm

10 cm

12 cm

D

C

B

A

11 cm

2 cm

6 cm

7 cm E

F


JPN PERAK Page 6

Worksheet 6 : Getting more interesting to solve triangles…..

Case 3: SSA

1. If two sides and a non-included angle [Case 3: SSA] of a triangle are given, then the

remaining side and angles are usually calculated by using a certain formula.

State and prove the formula.

2. Show how you can use that formula only to solve the following two triangles.

(a) (b)

3. The diagram below shows a quadrilateral ABCD. AEC is a sector of a circle with centre A.

Given that the perimeter of sector AEC is 1.5π + 18 cm, calculate the value of in radian.

A

C

B

75o

8 cm

6 cm 40

o

6 cm

5 cm

P Q

R

D

30o

10 cm

C B

A

7 cm

E 50

o


JPN PERAK Page 7

Worksheet 7 : Getting easier to find the areas of triangles…..

As easy as ABC

1. ½ × base × height is a formula that is used to calculate the area of a triangle given the base

and height of the triangle.

Derive a formula that can be used to calculate the area of a triangle given two sides and the

included angle of the triangle.

2. Show how you can use the formula to calculate the areas of triangles in each of the following

cases.

Case 1: SAS

(a) (b)

Case 2: SSS

Case 3: SSA

(a) Triangle ABC with the following measurements:

AB = 10 cm , AC = 7 cm and ACB = 135o

(b) Triangle PQR with the following measurements:

PQ = 12 cm , PR = 9 cm and PQR = 0.45 rad.

3. The diagram below shows a sector OAB with centre O and radius 16 cm. AM = 14 cm and

AOB = 3

1 π radian.

Calculate, in cm2, the area of the shaded region.

C B

A

10 cm

8 cm

30o

P

R Q

6 cm 8 cm

2.09 rad.

K

L M 6 cm

5 cm 4 cm

M B

A

O


JPN PERAK Page 8

Worksheet 8 : Getting more exciting – further challenges …..

1. Mission possible / impossible ?

Mission 1

Anita was asked to bend a piece of wire 19 cm long to form a triangle. The lengths of two sides of the

triangle must be 6 cm and 10 cm.

Mission 2

Amarjit was also asked to bend a piece of wire 19 cm long to form a triangle ABC such that

AB = 5cm, BC = 8 cm and ABC = 3

1 π rad.

Mission 3

Ah Chong was asked to construct a triangle PQR with an area of 36 cm2. The lengths of PQ and PR

must be respectively 8 cm and 6 cm.

Mission 4

The diagram below shows two roads. KL = 8 km.

An extra road, not more than 5 km long, is to be built from point L to a point M on the road KP.

Mission 5

A farmer would like to fence up a triangular region ABC as shown in the following diagram.

The farmer intends to use a 10 m fence and would like the area enclosed to be 6.5 m2.

Determine whether these five missions are possible or impossible to accomplish.

For Mission 5, use two methods.

40o

L

K M P

wall A B

C

150o


JPN PERAK Page 9


2. Triangles in beautiful progression

Progressions are beautiful number sequences.

Determine the condition for the three sides of a triangle to form

(a) an arithmetic progression,

(b) a geometric progression.

For each case, construct a triangle as an example.


JPN PERAK Page 10


3. Surprisingly different!

Arun and Abil each bought a piece of land. Each piece of land is in the shape of a quadrilateral ABCD

with the following measurements:

(a) Calculate, in m, the length of AC.

(b) Arun found out that the piece of land he bought was smaller than the piece of land that Abil

bought.

(i) Explain why this situation can occur.

Support your explanation by calculating the difference between the areas of the two

pieces of land.

(ii) Construct, using a scale of 1 cm to 10 m, the two pieces of land that Arun and Abil

bought.

AB = 50 m, AD = BC = 80 m

ABC = 60o, ADC = 30

o


JPN PERAK Page 11

Worksheet 11 : Feeling the joy of completing a project work…..

Sweet reflection!

It is a good practice to make a reflection on what you have done or achieved. This will enable you to

be more successful the next time you carry out a similar task.

Express creatively a sweet reflection on this project work that you have done.

Include in your reflection the benefits you get from this project work.

Did it in any way help you to become more prepared for the 2015 SPM examination?

perak state additional mathematics project workjpnperak.moe.gov.my/jpn/attachments/article/2728/item...

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