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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.
Perak State Additional Mathematics Project Work 2015
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
Perak State Additional Mathematics Project Work 2015
JPN PERAK Page 3
Worksheet 3 : Getting curious about triangles…..
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
Perak State Additional Mathematics Project Work 2015
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Worksheet 4 : Getting curious about triangles…..
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
Perak State Additional Mathematics Project Work 2015
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
Perak State Additional Mathematics Project Work 2015
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
Perak State Additional Mathematics Project Work 2015
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
Perak State Additional Mathematics Project Work 2015
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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
Perak State Additional Mathematics Project Work 2015
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Worksheet 9 : Getting more exciting – further challenges …..
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.
Perak State Additional Mathematics Project Work 2015
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Worksheet 10 : Getting more exciting – further challenges …..
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
Perak State Additional Mathematics Project Work 2015
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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?