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