My Additional Mathematics Modules - Form V - Solution of Triangles by nklye

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									                                         My
                          ADDITIONAL
                         MATHEMATICS
                                    Modules
                                        TOPIC 10
                                       (Version 2012)




                                SOLUTION
                                         of
                                TRIANGLES
                              ( PENYELESAIAN SEGITIGA )




                                            by:

                                          NgKL




edmet-nklpunya.blogspot.com
SOLUTION OF TRIANGLES                                                                         TOPIC 10

10.1      SINE RULE (PETUA SINUS )

          1.    Sine rule for any  ABC , ( Petua Sinus untuk sebarang  ABC )
                           a     b     c
                                   
                         Sin A Sin B Sin C

          2.    Sine rule can be expressed in reciprocal form. (Petua sinus boleh juga diungkapkan dalam
                bentuk salingan,)
                         Sin A          Sin B       Sin C
                                               
                            a            b            c



  Exercise 10.1 (Sine Rule / Petua Sinus )

1.     Calculate the length of AC.




2.     Determine ABC




3.     Find ACB




                                                            2
SOLUTION OF TRIANGLES                                                  TOPIC 10

4.   Calculate ABC




5.    In the diagram shown on the right side, ABC
     is a straight line. Calculate the length of
      (a) BC
      (b) AB




6.   In the diagram shown on the right side, PQR is a straight line.
     Find

     (a) RQS
     (b) The length of RS




7.   In the diagram shown on the right, ABC is a straight line.
     Calculate

     (a) ACD
     (b) CBD




                                                    3
SOLUTION OF TRIANGLES                                                 TOPIC 10




8.   Diagram on the right side shows PQR and PRS ,
     QRS is a straight line. Find

     (a) the length of PQ
     (b) RSP




9.   Diagram on the right side shows ABC
     and ACD , BCD is a straight line. Find

     (a) the length of AC
     (b) ABC




10. In the diagram shown on the right side, ABC is a straight line.
    Calculate

     (a) the length of CD
     (b) DAB if the AD = 13.3 cm




                                                  4
SOLUTION OF TRIANGLES                                                                                     TOPIC 10

10.2    Sine Rule For Ambigious Cases
        (Petua Sinus yang melibatkan Kes Berambuguiti)
        Ambiguity occurs when the length of two sides and a non-included acute angle are given. The
        non-included acute angle must be opposite the shortest side of the two sides given. In such
        case, there is a possibility of two forms of triangle occur as shown in the diagram below.
        (Kes berambuguiti berlaku apabila panjang dua sisi dan satu sudut tirus bukan kandung diberi. Sudut tirus bukan
        kandung itu bertentangan dengan sisi terpendek daripada dua sisi yang diberi. Dalam kes ini, ada kemungkinan
        terdapat dua buah segi tiga seperti yang ditunjukkan dalam rajah di bawah.)




        (a)    Based on the diagram, a < c and angle A is acute and non-included, side BC is shoter
               than side AB and is opposite the non-included acute angle A, then ambiguity can occur
                (Berdasarkan rajah di atas, a< c dan sudut A adalah tirus dan bukan sudut kandung kandung, sisi BC
               lebih pendek daripada sisi AB yang bertentangandengan sudut tirus bukan kandung A, maka kes
               berambuguiti akan berlaku.)

        (b)    In this case, the ambiguity means two forms of triange (with similar values of length
               of sides and non-included acute angle) can be formed; i.e. ABC and ABC' .
               ( Kes berambuguiti bermakna bahawa terdapat dua segitiga yang dapat dibentuk iaitu ABC dan
                ABC' .)
        (c)    Ambiguity would not be occurred if the angle given is an obtuse angle.
               ( Kes berambuguiti tidak akan berlaku jika sudut yang diberi ialah sudut cakah.)




Exercise 10.2 (Sine Rule For Ambigious Cases / Petua Sinus dengan Kes Berambuguiti)

1.      In a triangle PQR, PQ = 8.5 cm, PR = 6 cm and  PQR  28 , sketch and label the possible
        new triangle can be formed and find the two possible PRQ .




2.      In a triangle PQR, PQ = 10 cm, PR = 7 cm and  PQR  42 , sketch and label the possible
        new triangle can be formed and find the two possible PRQ .

                                                                          R


                                                                          7 cm
                                                                                                  42o
                                                                               P           10 cm               Q
                                                          5
SOLUTION OF TRIANGLES                                                                           TOPIC 10

2.   ABC is a triangle with AB = 8 cm, AC = 4 cm and  ABC  25 . Sketch and label the
     possible new triangle can be formed and find the two possible  ACB .

                                                                                C

                                                                 4 cm
                                                                                        25o
                                                               A                                        B
                                                                                    8 cm




3.   In a triangle XYZ, XY = 7 cm, XZ = 9 cm and XZY  48 . Sketch the possible new
     triagle can be formed. Find the two possible XYZ and the length of YZ


                                                          Y
                                                                                                Z
                                                                                        48o
                                                              7 cm
                                                                                        9 cm


                                                                     X


4.   Diagram shows JKL is a triangle with JK = 10 cm, JL = 15 cm and  JLK  32 .
     Sketch and label a new triangle formed on the triangle JKL with the length JL and JK and
     JLK remain the same. Find the possible JKL


                                                                    15 cm
                                        L             o
                                                    32
                                                                                                    J

                                                                                           10 cm

                                                                            K




5.   KLM is a triangle with KL = 15 cm, LM = 9.2 cm dan  MKL  36 . Sketch and label a new
     triangle formed on the triangle KLM with the length KL and LM and  MKL  36 remain
     the same. Find the possible  MLK .


                                                                                    M
                                                                                               9.2 cm

                                                              36o
                                                K                                                       L
                                                                         15 cm
                                            6
SOLUTION OF TRIANGLES                                                                TOPIC 10

10.2      COSINE RULE (PETUA KOSINUS).

          1.    Cosine rule for any  ABC , ( Petua Kosinus untuk sebarang  ABC )

                               a 2  b 2  c 2  2bc Cos A
                               b 2  a 2  c 2  2ac Cos B
                               c 2  a 2  b 2  2ab Cos C




 Exercise 10.3 (Cosine Rule / Petua Kosinus)


1.     Calculate the length of PQ.




2.     Determine the length of PQ




3.     Calculate the length of AB




                                                    7
SOLUTION OF TRIANGLES                          TOPIC 10

4.   Determine DEF




5.   Calculate the largest angle in XYZ




6.    Find    (i) PQS
              (ii) RSQ




7.    Find (i) ACD
           (ii) the length of AB




                                           8
SOLUTION OF TRIANGLES                                                                                       TOPIC 10


8.     In the diagram shown on the right side, BCD is
         a straight line. Determine

         (i) ACB
         (ii) the length of CD.




10.3          AREA OF A TRIANGLE (LUAS SEGITIGA)

          1.     If a triangle ABC have the measurements as shown in the diagram, then
                 ( Jika sebuah segitiga ABC mempunyai ukuran-ukuran seperti ditunjukkan dalam gambarajah,

                                       1                                                           A
                   Area Δ ABC             bc Sin A
                                       2
                                                                                         c                   b
                                                                                                   Ao
          .
                Similarly, area of the triangle can be determine by                                               C
                the following formulae;                                       B
                (Dengan cara yang sama, luas segitiga itu boleh ditentukan
                 dengan rumus berikut;)

                                      1                 1
                  Area Δ ABC              ac Sin B        ab Sin C
                                      2                 2




       Exercise 10.4 ( Area of a Triangle / Luas Segitiga )


1.     Calculate, correct to two decimal places, the area of the following triangles.


        (a)                                                             (b)




                                                              9
SOLUTION OF TRIANGLES                                                                TOPIC 10


      (c)                                          (d)




2.   Find the area of the triangle ABC if:


      (a) b = 29 cm
          c = 46 cm
          A = 3524




      (b) a = 7.2 cm
          c = 4.8 cm
          B = 42




3.      PQR is an acute triangle with PQ = 3 cm and QR = 6 cm. If the area of  PQR is 7 cm 2 ,
        find the value of angle R.


                                                                    R
                                                                                  6 cm

                                                                            Area = 7 cm2        Q

                                                                                         3 cm
                                                                              P




                                              10
SOLUTION OF TRIANGLES                                                               TOPIC 10

4.   In a  ABC, A  108 and AB  2 AC . Given that the area of the triangle ABC is 5 cm 2 ,
     determine the length of side AC.

                                                                              A

                                                                             108o

                                                                                            C
                                                          B




5.   Calculate the area of the triangle ABC as shown in
     the diagram on the right side.




7.   In the diagram shown on the right side, a crane ABC is picking up
     a load D. Given that BA is perpendicular to AE, find

     (a)    BAC
     (b)   Height of D from AE




                                              11
SOLUTION OF TRIANGLES                                        TOPIC 10


     Exercise 10.5 – Extra Mile

1.     In the diagram shown on the right hand side, BMC is
       a staright line. Find

       (a) the length of AC
       (b) the length of AB
       (c) area of triangle ABC




2.     In the diagram shown on the right hand side, find

       (a) the length of CD
       (b) the length of AD




3.     In the diagram shown on the right hand side, ABC
       and CDE are straight lines.

       (a) Find cos ACE
       (b) Hence, determine the length of AE.




                                                   12
SOLUTION OF TRIANGLES                                         TOPIC 10

4.   In the diagram shown on the right hand side, QRS
     is a straight line.
     Find
     (a)  PRS
     (b) the length of RS
     (c) Area of triangle PQS




5.   In the diagram shown on the right hand side, calculate

     (a) the length of AB,
     (b) sketchthe new triangle ABC when AC is extended,
         while the length of AB, BC and  BAC remain
         the same.
     (c) area of the new triangle ABC mentioned in (b).




                                                  13
SOLUTION OF TRIANGLES                                                           TOPIC 10

6.   The following diagram shows a triangle PQR.

     Calculate,
     (a)  PQR                                                                  [2 marks]
                                                             P


                                                                   16.41 cm
                                                    8.1 cm

                                                                              28o
                                                    Q                                   R



     (b) the length of QR.                                                      [3 marks]




     (c) the area of the triangle PQR.                                          [2 marks]




      (d) A triangle P’Q’R’ has the same measurements as those given for triangles PQR,
        that is,
          P’R’ = 16.41 cm, P’Q’ = 8.1 cm and  P’R’Q’ = 28o, but which is different in
        shape to
          triangle PQR.

          (i) Sketch the triangle P’Q’R’.                                       [2 marks]




          (ii) What is the size of  P’Q’R’?                                    [1 marks]




                                               14
SOLUTION OF TRIANGLES                                                              TOPIC 10

  PAST YEAR SPM QUESTIONS

1. 2007 / PAPER 2 − SECTION C / Q15

   Diagram 7 shows quadrilateral ABCD.
                                                                   A
  (a)    Calculate
                                                       5.6 cm
         (i) the length, in cm, of AC
         (ii)  BCD                     [4 marks]      B   105 o
                                                                           16.4 cm




                                                                                50 o

                                                                       C     6 cm      D

                                                                           Diagram 7


   (b)   Point A’ lies on AC such that A’B = AB.
          (i) Sketch Δ A’BC.




         (ii) Calculate the area, in cm2, of Δ A’BC.                   [6 marks]




                                                15
SOLUTION OF TRIANGLES                                                                 TOPIC 10

2. 2009 / PAPER 2 − SECTION C / Q13

    Diagram 12 shows a trapezium KLMN. KN is parallel to LM and  LMN is obtuse.

    Find
                                                                       N
    (a)    the length, in cm, of LN.    [2 marks]

                                                             12.5 cm



                                                         K     80 o

                                                                                           M
                                                                           32 o
                                                                                      5.6 cm
                                                                                  L

                                                                       Diagram 12


    (b)    the length, in cm, of MN.                                                   [3 marks]




.
    (c)     LMN.                                                                      [3 marks]




    (d)    the area, in cm2, of triangle LMN.                                          [2 marks]




                                                    16
SOLUTION OF TRIANGLES                                                                         TOPIC 10

3.   2005 / PAPER 2 / SECTION C / Q12

     Diagram 7 shows triangle ABC.

     (a)   Calculate the length, in cm, of AC.                                                 [2 marks]

                                                                                          A


                                                                      20 cm




                                                               B       65o


                                                                        15 cm
                                                                                          C
                                                                              Diagram 7


     (b)   A quadrilateral ABCD is now formed so that AC is a diagonal,  ACD = 40o and AD = 16 cm.
           Calculate the two possible values of  ADC
                                                                                             [2 marks]




     (c)   By using the acute  ADC from (b), calculate
           (i) the length, in cm, of CD,
           (ii) the area, in cm2, of the quadrilateral ABCD.                                   [6 marks]




                                                    17
SOLUTION OF TRIANGLES                                                                             TOPIC 10

4.   2004 / PAPER 2 / SECTION C / Q13

     Diagram 6 shows a quadrilateral ABCD such that  ABC is acute.
                                                                                    D
     (a) Calculate
                                                                 9.8 cm                  5.2 cm
         (i)  ABC
                                                                              12.3 cm              C
                                              A
                                                                  40.5o
                                                                                                       9.5 cm



                                                                     Diagram 6
                                                                                                           B

         (ii)  ADC




         (iii) the area, in cm2, of quadrilateral ABCD.                                       [8 marks]




     (b) A triangle A’B’C’ has the same measurements as those given for triangles ABC, that is,
         A’C’ = 12.3 cm, C’B’ = 9.5 cm and  B’A’C’ = 40.5o, but which is different in shape to triangle ABC.

         (i)    Sketch the triangle ABC.




         (ii)   State the size of  ABC.                                                      [2 marks]




                                                      18
SOLUTION OF TRIANGLES                                                                          TOPIC 10

5.   2006 / PAPER 2 / SECTION C / Q13

     Diagram 5 shows a quadrilateral ABCD. The area of triangle BCD is 13 cm2 and     BCD is acute.
     Calculate

     (a)    BCD                          [2 marks]
                                                                               D
                                                                                   5 cm

                                                                         40o              C

                                                                                      6 cm      Diagram 5

                                                                               B
                                                                  9 cm
                                                 A

     (b)   the length, in cm, of BD,                                                             [2 marks]




     (c)     ABD,                                                                               [3 marks]




     (d)   the area, in cm2, of the quadrilateral ABCD.                                         [3 marks]




                                                      19

								
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