Solution of Triangle No 1

Solutions of Triangles 1

SOLUTION OF TRIANGLES

1.2 Use 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.

(1) Diagram 1 shows the triangle ABC.

Calculate the length of BC.

Answer :

00 40sin

2.8

75sin=

BC

0

075sin

40sin

2.8×=BC

Using the scientific calculator,

BC = 12.32 cm

(2) Diagram 2 shows the triangle PQR

Calculate the length of PQ.

[ 8.794 cm ]

(3) Diagram 3 shows the triangle DEF.

Calculate the length of DE.

[ 10.00 cm ]

(4) Diagram 4 shows the triangle KLM.

Calculate the length of KM.

[ 11.26 cm ]

Diagram 2

D

E F

600 35

0 16’

15 cm

Diagram 3

L K

M

420

630

15 cm

Diagram 4

Diagram 1

To find unknown sides :

C

c

B

b

A

a

sinsinsin==

To find unknown angles :

c

C

b

B

a

A sinsinsin==



Calculate the length of AC.

Answer :

0000 657540180 =−−=∠ABC

00 40sin

2.8

65sin=

AC

0

065sin

40sin

2.8×=BC

Using the scientific calculator,

AC = 11.56 cm

(6) Diagram 6 shows the triangle PQR

Calculate the length of PR.

[ 6.527 cm ]


Calculate the length of EF.

[ 17.25 cm ]


Calculate the length of KL.

[ 16.26 cm ]

Diagram 5

Diagram 6

D

E F

600 35

0 16’

15 cm

Diagram 7

L K

M

420

630

15 cm

Diagram 8


Task 2 : Find the unknown sides of a triangle when two of its angles and the side not corresponding

to the angles are known.


Calculate the length of BC.

Answer :

0000 557847180 =−−=∠ABC

00 55sin

2.11

47sin=

BC

0

047sin

55sin

2.11×=BC

Using scientific calculator,

BC = 9.9996 cm or 10.00 cm


Calculate the length of AC.

[ 4.517 cm ]

(11) Diagram 11 shows the triangle PQR.

Calculate the length of PQ.

[ 3.810 cm ]


Calculate the length of DE.

[ 5.189 cm ]

Diagram 10

R P

Q

250

280

7.2 cm

Diagram 11

D

E F 72

0 51

0

5.6 cm

Diagram 12

Diagram 9


Task 3 : Find the unknown angles of a triangle when two sides and a non-included angle are given.


Find ∠ACB.

Answer :

15

60sin

10

sin 0

=C

15

60sin10sin

0

=C

5774.0sin =C

5774.0sin 1−=C

027.35=C

(2) Diagram 2 shows the triangle KLM

Find ∠KLM

[ 27.360 ]


Find ∠DFE.

[ 11.090 ]


Find ∠QPR.

[ 36.110 ]

D

E F

3.5 cm

430 24’

12.5 cm

Diagram 3

L K

M

9 cm

500

15 cm

Diagram 2

R P

Q

10 cm

1300

13 cm

Diagram 4

A

B C

600

15 cm

Diagram 1

10 cm



Find ∠ABC.

Answer :

14

110sin

9

sin 0

=A

14

110sin9sin

0

=A

6041.0sin =A

6041.0sin 1−=A

016.37=A

0

000

84.32

16.37110180

=

−−=∠ABC


Find ∠KLM.

[ 138.640 ]


Find ∠DFE.

[ 124.460 ]


Find ∠PQR.

[ 94.150 ]

L K

M

4.2 cm

250

2.8 cm Diagram 6

E D

F

340

6.7 cm

Diagram 7

4.4 cm

B

A

C

1100

14 cm

Diagram 5 9 cm

P

R Q

550

12.3 cm

Diagram 8

7.7 cm


Task 4 : Find the unknown side of a triangle when two sides and a non-included angle are given.


Given that ∠ACB is an obtuse angle, find

the length of AC.

Answer :

9

37sin

14

sin 0

=C

9

37sin14sin

0

=C

9362.0sin =C

9362.0sin 1−=C

058.110=C

0

000

42.32

3758.110180

=

−−=B

00 37sin

9

42.32sin=

AC

0

0

37sin

42.32sin9=AC

AC = 8.018 cm

(2) Diagram 2 shows the triangle KLM

Given that ∠KLM is an obtuse angle, find

the length of ML.

[ 2.952 cm ]

L K

M

7 cm

400

9 cm

Diagram 2

B

A

C

370 14 cm

Diagram 1 9 cm



Given that the value of ∠EDF is greater than

900, find the length of DE.

[ 5.040 cm ]


Given that ∠PQR is an angle in the second

quadrant of the cartesian plane, find the

length of QR.

[ 2.707 cm ]


Given that ∠KLM is an angle in the second

quadrant of the cartesian plane, find the

length of KL.

[ 9.686 cm ]

D

E F 11 cm

420

8 cm

Diagram 3

R P

Q

6.9 cm

460

8.5 cm

Diagram 4

L K

M

17.3 cm

230

9.2 cm

Diagram 5

Solution of Triangle No 1

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