solution of triangles
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SOLUTION OF TRIANGLES
SOLUTION OF TRIANGLES3.2Solve problems involving three-dimensional objects
Task : Answer all the questions below.ACGHDBFEG(1)
The diagram above shows a cuboid with a rectangular base, ABCD. Given that AB = 6cm, BC = 4cm and CG=3 cm.
FindBHCSolution:
HC=
BD =
HB =
= 0.8589
BHC = 30.810
DBFEHGCA(2)
The diagram above shows a cuboid with a rectangular base ABCD. Given that AB = 16 cm, BC = 4cm and CG=13 cm.
Find BHC
[10.98]
ABCGFEHD(3)
The diagram above shows a cuboid with a rectangular base ABCD. Given that AB = 6 cm, BC = 4cm and CG = 3 cm.
FindBGD.
[74.44]
ABCDEFGHThe diagram on the left shows a cuboid with a rectangular base ABCD. Given that DG=6.1 cm, BG=7.2 cm and BGD =41.02. Find the length of BD.4.
[ 4.772 cm ]
HGThe diagram on the left shows a cuboid with a rectangular base ABCD. Given that BC = 8.2 cm, CG = 6.42 cm, AB = 12.03 cm and ABG =110.02. Find the length of AG.5.
D
FEC
A
B
[ 17.91 cm ]
To find unknown sides :To find unknown angles :
10. SOLUTION OF TRIANGLES1.2Use Sine Rule to find the unknown sides or angles of a triangle.
Task 1: Find the unknown sides of a triangle when two of its angles and one of the corresponding sides are known. Diagram 1 (1) Diagram 1 shows the triangle ABC.
Calculate the length of BC.Answer :
Using the scientific calculator,
BC = 12.32 cm
Diagram 2(2) Diagram 2 shows the triangle PQR
Calculate the length of PQ.
[ 8.794 cm ]
DEF600350 1615 cmDiagram 3(3) Diagram 3 shows the triangle DEF.
Calculate the length of DE.
[ 10.00 cm ]
LKM42063015 cmDiagram 4(4) Diagram 4 shows the triangle KLM.
Calculate the length of KM.
[ 11.26 cm ]
Diagram 5 (5) Diagram 5 shows the triangle ABC.
Calculate the length of AC.Answer :
Using the scientific calculator,
AC = 11.56 cm
Diagram 6(6) Diagram 6 shows the triangle PQR
Calculate the length of PR.
[ 6.527 cm ]
DEF600350 1615 cmDiagram 7(7) Diagram 7 shows the triangle DEF.
Calculate the length of EF.
[ 17.25 cm ]
LKM42063015 cmDiagram 8(8) Diagram 8 shows the triangle KLM.
Calculate the length of KL.
[ 16.26 cm ]
Task 2: Find the unknown sides of a triangle when two of its angles and the side not corresponding to the angles are known.Diagram 9(9) Diagram 9 shows the triangle ABC.
Calculate the length of BC.Answer :
Using scientific calculator,
BC = 9.9996 cm or 10.00 cm
Diagram 10(10) Diagram 10 shows the triangle ABC.
Calculate the length of AC.
[ 4.517 cm ]
(11) Diagram 11 shows the triangle PQR.
RPQ2502807.2 cmDiagram 11
Calculate the length of PQ.
[ 3.810 cm ]
(12) Diagram 12 shows the triangle DEF.
DEF720510 5.6 cmDiagram 12
Calculate the length of DE.
[ 5.189 cm ]
Task 3 : Find the unknown angles of a triangle when two sides and a non-included angle are given.ABC60015 cmDiagram 110 cm(1) Diagram 1 shows the triangle ABC.
Find ACB.Answer :
LKM9 cm 50015 cmDiagram 2(2) Diagram 2 shows the triangle KLM
Find KLM
[ 27.360 ]
DEF3.5 cm430 2412.5 cmDiagram 3(3) Diagram 3 shows the triangle DEF.
Find DFE.
[ 11.090 ]
RPQ10 cm 130013 cmDiagram 4(4) Diagram 4 shows the triangle PQR.
Find QPR.
[ 36.110 ]
BAC110014 cmDiagram 59 cm(5) Diagram 5 shows the triangle ABC.
Find ABC.
Answer :
LKM4.2 cm2502.8 cmDiagram 6(6) Diagram 6 shows the triangle KLM.
Find KLM.
[ 138.640 ]
EDF3406.7 cmDiagram 74.4 cm(7) Diagram 7 shows the triangle DEF.
Find DFE.
[ 124.460 ]
PRQ55012.3 cmDiagram 87.7 cm(8) Diagram 8 shows the triangle PQR.
Find PQR.
[ 94.150 ]
Task 4: Find the unknown side of a triangle when two sides and a non-included angle are given.BAC37014 cmDiagram 19 cm(1) Diagram 1 shows the triangle ABC.
Given that ACB is an obtuse angle, find the length of AC.
Answer :
AC = 8.018 cm
LKM7 cm 4009 cmDiagram 2(2) Diagram 2 shows the triangle KLM
Given that KLM is an obtuse angle, find the length of ML.
[ 2.952 cm ]
DEF11 cm420 8 cmDiagram 3(3) Diagram 3 shows the triangle DEF.
Given that the value of EDF is greater than 900, find the length of DE.
[ 5.040 cm ]
RPQ6.9 cm 4608.5 cmDiagram 4(4) Diagram 4 shows the triangle PQR.
Given that PQR is an angle in the second quadrant of the cartesian plane, find the length of QR.
[ 2.707 cm ]
LKM17.3 cm2309.2 cmDiagram 5(5) Diagram 5 shows the triangle KLM.
Given that KLM is an angle in the second quadrant of the cartesian plane, find the length of KL.
[ 9.686 cm ]
10. SOLUTION OF TRIANGLES2.2Use Cosine Rule to find the unknown sides or angles of a triangle.
Task 1: Find the unknown side of a triangle when two sides and an included angle are given.(1) Diagram 1 shows the triangle PQR such that PR =12.3 cm , QR =16.4 cm and
PRQ = 67 .
Diagram 112.3 cmP16.4 cmx cmQR670
Find the value of x.Solution :
= 262.1
(2) Diagram 2 shows the triangle PQR such that
PQ =7 cm, QR =5 cm and PQR = 75 .
5cmx cm7 cmPQR7505 cmDiagram 2
Find the value of x.
[ 7.475 ]
(3) Diagram 3 shows a triangle with sides 5 cm , 13 cm and an included angle 43 .
x cmE13 cm430 5 cmDiagram 3
Find the value of x .
[ 9.946 ]
CAB6.3 cm 5307 cmDiagram 4(4) Diagram 4 shows the triangle PQR.
Find the length of BC.
[ 5.967 cm ]
(5) Diagram 5 shows the triangle KLM.
LKM4 cm4805.8 cmDiagram 5
Find the length of LM.
[ 4.312 cm ]
(6) Diagram 6 shows the triangle PQR.
750 312.23 cm 5.40 cmPQRDiagram 6
Find the length of PR.
[ 5.302 cm ]
(7) Diagram 7 shows a triangle with sides 6.21 cm , 10.51 cm and an included angle 360 39 .
x cm10.51cm360 396.21cmDiagram 7
Find the value of x .
[ 6.656 ]
Task 2: Find the unknown angle of a triangle when three sides are given.(1) In Diagram 1, ABC is a triangle where AB = 13 cm, AC = 14 cm and BC= 15 cm.
CAB13cm14 cm15 cmDiagram 1
Find .Solution :
=0.3846
(2) Diagram 2 shows a triangle ABC where AB = 11 cm, AC = 13 cm and BC= 16 cm.
CAB11cm13 cm16 cmDiagram 2
Find .
[ 83.17]
(3) Diagram 3 shows a triangle ABC where AB = 13 cm, AC = 16 cm and BC = 17.5 cm.
CAB13cm16 cm17.5 cmDiagram 3
Calculate
[ 73.41]
(4) Diagram 4 shows a triangle ABC where AB = 12.67 cm, AC = 16.78 cm and
AB12.67cm16.78 cm19.97 cmDiagram 4C BC= 19.97 cm.
Calculate
[39.17]
(5) In Diagram 5, PQR is a triangle such that PR = 6.45 cm, RQ = 2.23 cm and
2.23 cm 6.45 cm5.40 cmPQRDiagram 5 PQ = 5.40 cm.
Find .
[108.07]
(6) In Diagram 6, PQR is a triangle such that PR = 23.5 cm, RQ = 12.5 cm and
12.5 cm 23.5 cm18.7 cmPQRDiagram 6 PQ= 18.7 cm.
Calculate the smallest angle in the triangle.
[31.96]
(7) For triangle ABC in Diagram 7, AB = 8.56 cm, AC = 11.23 cm and BC= 14.51 cm.
CAB8.56cm11.23cm14.5 1cmDiagram 7
Calculate the largest angle in the triangle.
[93.33]
(8) For triangle ABC in Diagram 8, AB = 13 cm, AC = 16 cm and BC= 17.5 cm.
CAB13cm16 cm17.5 cmDiagram 8
Calculate the second largest angle in the triangle.
[61.19]
Area of = = = 10. SOLUTION OF TRIANGLES
3.1Use the formula or its equivalent to find the area of a triangle.
Task : Find the area of a triangles given in each of the following..(1) In Diagram 1, ABC is a triangle with
AB= 6 cm, AC = 9 cm and .
Diagram 1
Find the area of ABCSolution:
Area of = 21.56 cm2
(2) In Diagram 2, ABC is a triangle with
AC= 6 cm, BC = 5 cm and . Diagram 2
Find the area of ABC.
[ 14.67 cm2 ]
(3) In Diagram 3, ABC is a triangle with
AC= 6 cm, BC = 8 cm and . Diagram 3
Find the area of ABC.
[ 20.78 cm2 ]
(4) In Diagram 4, ABC is a triangle with AC= 6 cm, BC = 12.5 cm and the reflex
angle.
Diagram 4
Find the area of ABC.
[ 35.24 cm2 ]
(5) In Diagram 5, ABC is a triangle such that AB= 12.5 cm , AC = 6 cm and ACB=80. Diagram 5
Find (a) CBA, (b) the area of the triangle.
Solution:
(a)
sin= = 0.4727
=sin -1 (0.2727) =28.21
(b) =71.79
Area of ABC= =35.62 cm2
(6) In Diagram 6, ABC is a triangle such that AB= 11 cm , AC = 15 cm and ACB=4534.
Diagram 6
Find (a) CBA, (b) the area of the triangle.
[ (a) 76.830 (b) 69.66 cm2 ]
(7) In Diagram 7, ABC is a triangle such that AC = 7 cm, AB = 15 cm and ACB = 11530. Diagram 7
Find (a) CBA, (b) the area of the triangle
[ (a) 24.910 (b) 33.46 cm2 ]
(8) In Diagram 8, ABC is a triangle where AB= 15 cm, BC =11 cm and AC=8 cm.
Diagram 8
Find (a) the smallest angle,
(b) the area of ABC.Solution
(a) =0.8545 B = 31 30
(b) Area of ABC = = 42.86
(9) In Diagram 9, ABC is a triangle where AB= 30 cm, BC =25 cm and AC=20 cm. Diagram 9
Find (a) the largest angle,
(b) the area of ABC.
[ (a) 82.820 (b) 248.04 cm2 ]
(10) In Diagram 10, ABC is a triangle where AB = 13 cm, AC = 14 cm and BC= 15 cm.
CAB13cm14 cm15 cmDiagram 10
Find (a) the second largest angle,
(b) the area of ABC.
[ (a) 59.490 (b) 84.00 cm2 ]
10. SOLUTION OF TRIANGLESFurther Practice with questions based on SPM format.
Task : Answer all the questions below.(1) Diagram 1 shows a trapezium LMNO. Diagram 1
Calculate (a) LNM, (b) the length of LN, (c) the area of OLN.
[ (a) 24.740 (b) 25.67 cm (c) 118.99 cm2 ]
(2) In Diagram 2, BCD is a straight line. Diagram 2
Find (a) ACD, (b) the length of BC, (c) the area of triangle ABD.
[ (a) 111.800 (b) 3.769 cm (c) 28.50 cm2 ]
(3) In Diagram 3, FGH is a straight line and G is the midpoint of FH. Diagram 3
Find (a) EFG, (b) the length of EG, (c) the area of triangle EGH.
[ (a) 52.620 (b) 11.23 cm (c) 52.62 cm2 ]
(4) Diagram 4 shows a quadrilateral KLNM. Diagram 4
Calculate (a) the length of LM, (b) MNL, (c) the area of quadrilateral KLNM.
[ (a) 12.92 cm (b) 31.730 (c) 141.65 cm2 ]
(5) In Diagram 5, QRS is a straight line. Diagram 5
Find (a) QPR, (b) the length of RS, (c) the area of triangle PRS.
[ (a) 54.310 (b) 4.157 cm (c) 74.75 cm2 ]
(6) In Diagram 6, BCD is a straight line. Diagram 6 Calculate (a) the length of AB, (b) CAD, (c) the area of triangle ACD.
[ (a) 6.678 cm (b)84.740 (c) 13.17 cm2 ]
Past year questions
1. SPM 2003 P2Q 15
Diagram below shows a tent VABC in the shape of a pyramid with triangle ABC as the horizontal base. V is the vertex of the tent and the angle between the inclined plane VBC and the base is
Given that VB =VC =2.2 m and AB =AC =2.6m, calculate(a) the length of BC if the area of the base is 3 m2. [3 marks](b) the length of AV if the angle between AV and the base is 250. [3 marks](c) the area of triangle VAB [4 marks][ANSWERS; 2.700, 3.149, 2.829 ]
2. SPM 2004 P2 Q13
Diagram below shows a quadrilateral ABCD such that is acute.
(a)Calculate (i) (ii) (iii) the area, in cm2, of quadrilateral ABCD [ 8 marks]
(b) A triangle ABC has the same measurements as those given for triangle ABC, that is, AC=12.3 cm, CB=9.5cm and , but which is different in shape to triangle ABC.(i) etch the triangle ABC
(ii) State the size of [2 marks]
[ANSWERS ; 57.23, 106.07, 80.96, 122.77]
3. SPM 2005 P2Q12 Diagram below shows triangle ABC
(a) Calculate the length, in cm, of AC. [2 marks](b) A quadrilateral ABCD is now formed so that AC is a diagonal, and AD =16 cm. Calculate the two possible values of [2 marks]( c) By using the acute from (b) , calculatethe length , in cm, of CD(ii) the area, in cm2, of the quadrilateral ABCD [6 marks]
[ANSWERS; 19.27, 50.73, 24.89, 290.1 ]
4. SPM 2006 P2 Q 13
The area of triangle BCD is 13 cm2 and is acute. Calculate(a) , [2 marks](b) the length, in cm, of BD, [ 2 marks](c) [3 marks](d) the area, in cm2, quadrilateral ABCD [3 marks] Diagram below shows a quadrilateral ABCD.
[ANSWERS ; 60.07, 5.573, 116.55 35.43 ]
5. SPM 2007 P2Q15
Diagram shows quadrilateral ABCD.(a) Calculate (i) the length, in cm, of AC (ii) [4 marks](b) Point A lies on AC such that AB =AB.(i) Sketch (ii) Calculate the area , in cm2 , of [6marks]
[ANSWERS 13.36, 23.88 13.80]
6. SPM 2008 P2Q14 In the diagram below, ABC is a triangle. ADFB,AEC and BGC are straight lines. The straight line FG is perpendicular to BC.
It is given that BD = 19 cm, DA =16 cm, AE = 14 cm,and (a) calculate the length, in cm, of(i) DE(ii) EC [5 marks](b) The area of triangle DAE is twice the area of triangle FBG,Calculate the length , in cm, of BG [4 marks](c) Sketch triangle ABC which has a different shape from triangle ABC [ 1 mark][ANSWERS: 19.344, 16.213, 10.502]
152Solutions of Triangles