Republic of the Philippines
Department of Education
Regional Office IX, Zamboanga Peninsula
9
Zest for Progress
Z P
eal of
4th QUARTER – Module 8:
SOLVING PROBLEMS INVOLVING
OBLIQUE TRIANGLES
Name of Learner:
___________________________
Grade & Section:
___________________________
Name of School:
___________________________
artnership
Mathematics – Grade 9
Alternative Delivery Mode
Quarter 4 - Module 8: Solving Problems Involving Oblique Triangles.
First Edition, 2020
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over them.
Published by the Department of Education
Secretary: Leonor Magtolis Briones
Undersecretary: Diosdado M. San Antonio
Development Team of the Module
Writer:
Genaro V. Caluyo, Jr.
Editors:
Mary Rose A. Castillo
Shirly V. Gajilomo
Illustrator:
Erwin G. Deloria
Ma. Mengielyn G. Cuartocruz
Reviewers: EPS, Mathematics
Vilma A. Brown, Ed. D.
Principal
Mujim U. Abdurahim
Management Team: SDS
Roy C. Tuballa, EMD, JD, CESO VI
ASDS
Jay S. Montealto, CESO VI
ASDS
Norma T. Francisco, DM, CESE
EPS Mathematics
Vilma A. Brown, Ed. D.
EPS LRMS
Aida F. Coyme, Ed. D.
Printed in the Philippines
Department of Education – Region IX, Zamboanga Peninsula
Office Address:
Tiguma, Airport Road, Pagadian City
Telefax:
(062) – 215 – 3751; 991 – 5975
E-mail Address:
region9@deped.gov.ph
1
Introductory Message
This Self – Learning Module (SLM) is prepared so that you, our dear learners, can continue
your studies and learn while at home. Activities, questions, directions, exercises, and
discussions are carefully stated for you to understand each lesson.
Each SLM is composed of different parts. Each part shall guide you step-by-step as you
discover and understand the lesson prepared for you.
Pre-tests are provided to measure your prior knowledge on lessons in each SLM. This will tell
you if you can proceed on completing this module or if you need to ask your facilitator or your
teacher’s assistance for better understanding of the lesson. At the end of each module, you
need to answer the post-test to self-check your learning. Answer keys are provided for each
activity and test. We trust that you will be honest in using these.
In addition to the material in the main text, notes to the Teacher are also provided to our
facilitators and parents for strategies and reminders on how they can best help you on your
home-based learning.
Please use this module with care. Do not put unnecessary marks on any part of this SLM. Use
a separate sheet of paper in answering the exercises and tests. Read the instructions carefully
before performing each task.
If you have any questions in using this SLM or any difficulty in answering the tasks in this
module, do not hesitate to consult your teacher or facilitator.
Thank you.
What I Need to Know
This learning material is designed for you to understand better how to
solvesproblems involving oblique triangles. The lesson is simplified supported with the
application of the law of sine and law of cosine, examples and exercises are presented
systematically.
It aims to cater the academic needs of diverse learners in achieving and
improving the twin goals of the Enhanced Basic Education Program or the K to 12
program ensuring the learners to become critical thinkers and problem solvers.
The language used recognizes the vocabulary level of grade 9 students. The lessons
follow developmentally sequenced teaching and learning processes to meet the
curriculum requirement.
After going through with this module, you are expected to solve problems
involving oblique triangles. M9GE-IVh-j-1.
Believe that learning can continue amidst the health crisis. Good luck, stay safe,
and God bless.
2
What I Know
Directions: Choose the letter that corresponds to your answer. Write your answer on
a separate sheet of paper.
For Questions 1-4. Refer to the problem below.
Sarah is planning to make a triangular garden. She wants to build a fence around
the garden to keep out the animals. The length of one side of the garden is 26 feet. If
the angles at the end of this side measures 780 and 440, find the following:
1. What is the measure of the third angle?
a.580
b. 680
c. 1000
2. What is the measure of the shorter side?
a. 21.30 feet
b. 36.23 feet
c. 15.23 feet
d. 20.12 feet
3. How about the longest side?
a. 29.99 feet
b. 39.34 feet
c. 50.55 feet
d. 40.45 feet
d. 500
4. What is the total length of fence needed to enclose the garden?
a. 107 feet
b. 47.34 feet
c. 77.29 feet
d. 97.56 feet
5. ∆ABC’s information below show which case/s of oblique triangle?
I. ASA II. SSS
III. SSA
a. I and III
b. II and III
c. III only
d. I only
B
17
13
A
270
C
6. The following are laws of cosine except:
a.
𝑠𝑖𝑛𝐴
𝑎
2
=
𝑠𝑖𝑛𝐵
𝑏
=
𝑠𝑖𝑛𝐶
c. 𝑐 2 = 𝑎2 + 𝑏2 − 2𝑎𝑏 cos 𝐶
𝑐
b. 𝑎 = 𝑏2 + 𝑐 2 − 2𝑏𝑐 cos 𝐴
d. 𝑏2 = 𝑎2 + 𝑐 2 − 2𝑎𝑐 cos 𝐵
For Number 7-10. Consider the problem below.
In ∆ ABC, if b = 38 cm, m  B = 46 and m A = 79.
C
b=38 cm
7. How long is side a?
a. 20.25 cm
b. 51.86 cm
c. 60.79 cm
d. 30.15 cm
a
790
460
B
A
c
3
8. How big is angle C?
a. 1700
c. 350
d. 550
9. What is the length of side c?
a. 43.27 cm
b. 70.85 cm
c. 51.20 cm
d. 35.56 cm
10. What is the perimeter of triangle ABC?
a. 51.86 cm
b. 43.27 cm
c. 38 cm
d. 133.13 cm
LESSON
1
b. 1050
SOLVING PROBLEMS INVOLVING
OBLIQUE TRIANGLES
What’s In
A. Directions: Identify the following triangles whether its oblique triangle or not.
1.
2.
3.
4.
_____________
___________
B.
C.
______________
___________
B. Directions: In your previous lessons you have learned about the laws of sine and
cosine.
Law of Cosine
𝑎 = 𝑏2 + 𝑐 2 − 2𝑏𝑐 cos 𝐴
𝑏2 = 𝑎2 + 𝑐 2 − 2𝑎𝑐 cos 𝐵
𝑐 2 = 𝑎2 + 𝑏2 − 2𝑎𝑏 cos 𝐶
Law of Sine
𝑠𝑖𝑛𝐴 𝑠𝑖𝑛𝐵 𝑠𝑖𝑛𝐶
=
=
𝑎
𝑏
𝑐
2
Solve triangle ABC illustrated at the right.
a = 4cm, A = 340, B = 920
Find:
1. Angle C
2. Side b
3. Side c
b=?
92
0
34
0
C=
4
a= 4 cm
ILLUSTRATIONS
mA + mB +m C = 180o
34o + 92o+ C = 1800
C = 180o – (34o + 92o)
C = 1800 - 1260
mC = 54
REASONS
The sum of the interior angles of a triangle is 1800
ILLUSTRATIONS
a
b
=
sin A sin B
bSin A = aSin B
a sin B
b=
sin A
4𝑆𝑖𝑛 920
𝑏=
𝑆𝑖𝑛34
4(0.99939)
b= 0.55919
REASONS
b = 7.14 cm
ILLUSTRATION
a
c
=
sin A sin C
cSin A = aSin C
𝑎 𝑠𝑖𝑛𝐶
𝑐=
𝑆𝑖𝑛 𝐴
4𝑆𝑖𝑛 530
𝑐=
𝑆𝑖𝑛34
4(0.79863)
c= 0.55919
c = 5.71 cm
Substitute the values of  A and  B
Apply Addition Property of Equality
Simplify
Measure of Angle C
Apply Law of Sine formula since the measures of
one side and 2 angles are given (SAA)
Substitute the measure of < B & < A.
Simplify
Length of side b
REASONS
Apply the Law of Sine formula since the measures
of one side and 2 angles are given (SAA)
Substitute the a=4, m < C & m < A.
Use your Calculator to find the value of sin 530 and
sin 320
Length of side c
5
A. Find a, m C if c = 8cm, b =10, and m  A =60
a=?
c=8cm
A
600
C
b=10cm
ILLUSTRATIONS
= + c2 - 2bc Cos A
a2 = 102 + 82 - 2(10)(8) Cos 600
a2
b2
REASONS
Law of Cosine
Substitute the value of b = 10, c = 8,
m<A = 60
a2 = 100 + 64 - 160 (0.5)
a2 = 164 - 80
Simplify
a2 = 84
a = √84
a = 9.2cm
Subtraction
Extract the square root
The length of side a
c2=a2+b2 – 2ab cos C
(8)2= (9.2)2 + (10)2 – 2(9.2)(10)cos C
64 =84.64 + 100 – 184 cos C
64-84.64-100 = -184 cos C
-120.64 = -184 cos C
-184
-184
cos C = 0.6556521739
Law of cosine
Substitution
Simplify
Combine like terms
Multiplicative inverse (Divide both sides of the
equation by -184
Quotient
Use scientific calculator get the value of angle C,
press shift Cosine
 C = 49.030
What’s New
ACTIVITY
PLEASE FIND ME!
Directions: Below are sketches depicting real-life situations. Find the missing parts
indicated in each sketch.
6
QUESTIONS:
1. What are the measures of the two base angles?
2. What is the length of the third side?
What is It
Knowledge or background of oblique triangle is very useful in solving real-life
situations but in solving it you need the following steps:
Step 1. Read and understand the problem
Step 2. Sketch the picture of the problem
Step 3. Identify the given data or cases
Step 4. Apply the appropriate law (sine or cosine) in solving the problem
1. A diagonal of a parallelogram is 26 cm. Find the perimeter of the parallelogram if
the angles between the sides and the diagonal are 350 and 400.
SOLUTION: to solve this problem, you need to figure out the problem by
drawing or sketching.
Ѳ
26cm
400
350
350
400
y
Ѳ
x
To solve for third angle:
Ѳ = 1800 – (350 + 400)
= 1800 – 750
Ѳ = 1050
To find x, Use the Law of sine
𝑥
𝑆𝑖𝑛 400
26
= 𝑆𝑖𝑛 1040
𝑥(𝑠𝑖𝑛104°)
𝑠𝑖𝑛104°
=
26(𝑠𝑖𝑛40°)
𝑠𝑖𝑛104°
𝒙 = 𝟏𝟕. 𝟓𝟓𝒄𝒎
7
To find y:
𝑦
𝑆𝑖𝑛 350
26
= 𝑆𝑖𝑛 1040
𝑦(𝑠𝑖𝑛104°)
𝑠𝑖𝑛104°
=
26(𝑠𝑖𝑛35°)
𝑠𝑖𝑛104
°
𝒚 = 𝟏𝟓. 𝟒𝟐 𝒄𝒎
The perimeter (P) of the parallelogram is
P = 2 (x + y)
P = 2 (17.55 cm + 15.42 cm)
P = 2( 32.97cm)
P = 65.94 cm
2. SURVEYING
To find the distance between two points A and B that are on opposite sides of a
river, a surveyor measures the distance to point C on the same side of the river as
point A. The distance from A to C is 24 feet. He then measures the angle from A
to B as 620 and measures the angle from C to B as 550. Find the distance from A
to B.
SOLUTION: The given data are two angles and one side (ASA), therefore this problem
can be solved using the law of sine.
𝑆𝑖𝑛 𝐴
𝑎
=
𝑆𝑖𝑛 𝐵
𝑏
𝑠𝑖𝑛63° 𝑠𝑖𝑛55°
=
24
𝐴𝐵
24(𝑠𝑖𝑛55°)
𝐴𝐵 =
𝑠𝑖𝑛63°
24(0.81915)
𝐴𝐵 =
0.89100
𝐴𝐵 = 22.06 𝑓𝑒𝑒𝑡
8
What’s More
ACTIVITY
SHOW ME YOUR SOLUTION
A balloon is sighted from two points on level ground. From point A, the angle of
elevation is 18° and from point B the angle of elevation is 12°. A and B are 8.5 miles
apart. Find the height of the balloon if a) A and B are on opposite sides of the balloon
and if b) A and B are on the same sides of the balloon.
MY TURN
YOUR TURN
Solution for a:
𝑠𝑖𝑛18° 𝑠𝑖𝑛150°
=
𝑎
8.5
8.5(𝑠𝑖𝑛18°)
𝑎=
𝑠𝑖𝑛150°
𝑎 = 5.25 𝑚𝑖𝑙𝑒𝑠
To find h, use 𝒔𝒊𝒏𝜽 =
For b show your solution here
𝒐𝒑𝒑𝒐𝒔𝒊𝒕𝒆
𝒉𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆
ℎ
5.25
ℎ = 5.25(𝑠𝑖𝑛12°)
ℎ = 1.09 𝑚𝑖𝑙𝑒𝑠
𝑠𝑖𝑛12° =
What I Have Learned
KEYPOINTS
•
An oblique triangle is a triangle which does not contain any right angle
•
Oblique triangle may be classified into two kinds of triangle namely: acute and
obtuse.
9
•
An acute triangle is a triangle whose angles are all less than 900.
•
An obtuse triangle is a triangle with one obtuse angle or one angle measure more
than 900.
•
The Law of Sines states that “In any triangle ABC, with a, b, and c as its sides, and
A, B and C as its angles.”
𝑠𝑖𝑛𝐴 𝑠𝑖𝑛𝐵 𝑠𝑖𝑛𝐶
=
=
𝑎
𝑏
𝑐
•
The Law of Cosine can be used to find the missing measures in a triangle if you
know the measures of:
1. Two sides and their included angle (SAS) and
2. Three sides (SSS)
𝑎2 = 𝑏2 + 𝑐 2 − 2𝑏𝑐 cos 𝐴
𝑏2 = 𝑎2 + 𝑐 2 − 2𝑎𝑐 cos 𝐵
𝑐 2 = 𝑎2 + 𝑏2 − 2𝑎𝑏 cos 𝐶
•
Steps in Solving Problem
Step 1. Read and understand the problem
Step 2. Sketch the picture of the problem
Step 3. Identify the given data or cases
Step 4. Apply the appropriate law (sine or cosine) in solving the problem
Directions: Briefly answer the questions below.
1. In what the case/s of oblique triangle you can apply the law of sine?
2. How about the Law of Cosine? In what case/s of oblique triangle can you apply this
law?
What I Can Do
Directions: For each of the given situational problems:
1. sketch the required triangle to solve the problem
2. use the appropriate law (Sine Law or Cosine Law) to solve it
PROBLEMS
1. Macario has three sticks measuring 19 cm, 23 cm, and 27 cm. He lays them down
to form a triangle. Find the measure of the angle formed by the 19-cm and 23-cm
sides to the nearest degree.
10
2. A triangular parcel of land with points A, B, C was to
be fenced. No data for the lengths of sides AB and BC
are available as shown in the figure below. How many
meters of fencing materials are needed to cover the
lot?
Assessment
Directions: Read each problem carefully then write only the letter of the correct
answer on a separate sheet of paper.
For Numbers 1-3, refer to the problem below.
The sides of a triangle ABC have lengths of 5.9 m, 2.8 m and 5.9 m. Find the
measure of the 3 angles.
1. What is the measure of the smallest angle?
a. 270
b. 460
c. 650
d. 170
2. What is the measure of the larger angle?
a. 670
b. 550
c. 760
d. 1050
3. How about the largest angle?
a. 1200
b. 900
c. 360
d. 770
For Numbers 4 – 6, refer to the problem below
A sign at a mountain overlook indicates that a hiker is 5.9.km from a microwave
tower M and 7.8 km from the highest visible peak P. The hiker estimates the angle
between M and P from her position to be 400.
4. How far is the microwave tower M to the highest visible peak P?
a. 5.01 km
b. 10.2 km c. 15.03 km d. 3.5 km
5. What is mHMP to the nearest degree?
a. 91
b.110
c. 69
d. 88
6. How about the measure of m  MPH ?
a. 36
b. 139
c. 49
d. 19
11
For Numbers 7-10. Refer to the problem below
A ship is sighted from two lighthouses that are 30 km apart on a shore. The
angle at the first lighthouse between the shore and the ship is 35 0. The angle at the
second lighthouse is 460.
7. How far is the ship from the second lighthouse?
a. 17.42 km
b. 27.23 km
c. 35 km
d. 25 km
8. How far is the ship from the first lighthouse?
a. 41.14 km
b. 21.85 km
c. 12.78 km d. 50 km
9. How large is the angle formed from the ship towards the two lighthouses?
a. 450
b. 1200
c. 990
d. 1000
10. What is the perimeter of the triangle?
a. 56 km
b. 99 km
c. 79.77 km d. 69.27 km
12
13
Assessment:
1. a
6. c
2. c
7. a
3. d
8. b
4. a
9. c
5. a
10. d
What I Can Do:
What I Have Learned:
1. If the triangle is not right, consider the law of sines. The law of sine can be applied
or used if the given case or cases are ASA, AAS and SSA.
2. If the law of sine is not helpful, use the law of cosines. The law of cosines is most
directly applicable if the case or cases given are SAS and SSS.
What’s More:
Answer for b
a=16.91 miles
h = 5.23 miles
What’s New:
1.
The measures of the two base angles are 47.50
2.
The length of the third side is 216.19 ft
What I Know:
1.
A 2. A
3. A
4. C
5. C
6. A
7. B
8. D
9. A
10. D
Answer Key
References:
Textbook:
Bryant, M. L., Bulalayao, L. E., Callanta, M. M., Cruz, J. D., De Vera, R. F., Garcia, G.
T., … Saladino, R. H. A. (2014). Mathematics Learner's Mate rial 9. (D. M. B. Versoza,
Ed.) (1st ed.). Pasig: Department of Education.
Orines, F. B., Esparrago, M. S., & Reyes, N. V. (2008). Advanced Algerbra,
Trigonometry, and Statistics (2nd ed.). Quezon: Phoenix Publishing
House,
Inc.
Videos:
Deriving the Law of Cosines. Retrieved June 15, 2020, from
https://www.youtube.com/watch?v=BZj6LmBKhmQ
14
I AM A FILIPINO
by Carlos P. Romulo
I am a Filipino – inheritor of a glorious past, hostage to the
uncertain future. As such, I must prove equal to a two-fold
task – the task of meeting my responsibility to the past, and
the task of performing my obligation to the future.
I am sprung from a hardy race – child many generations
removed of ancient Malayan pioneers. Across the centuries,
the memory comes rushing back to me: of brown-skinned
men putting out to sea in ships that were as frail as their hearts
were stout. Over the sea I see them come, borne upon the
billowing wave and the whistling wind, carried upon the
mighty swell of hope – hope in the free abundance of the new
land that was to be their home and their children’s forever.
This is the land they sought and found. Every inch of shore
that their eyes first set upon, every hill and mountain that
beckoned to them with a green and purple invitation, every
mile of rolling plain that their view encompassed, every river
and lake that promised a plentiful living and the fruitfulness
of commerce, is a hollowed spot to me.
By the strength of their hearts and hands, by every right of
law, human and divine, this land and all the appurtenances
thereof – the black and fertile soil, the seas and lakes and
rivers teeming with fish, the forests with their inexhaustible
wealth in wild and timber, the mountains with their bowels
swollen with minerals – the whole of this rich and happy land
has been for centuries without number, the land of my
fathers. This land I received in trust from them, and in trust
will pass it to my children, and so on until the world is no
more.
I am a Filipino. In my blood runs the immortal seed of heroes
– seed that flowered down the centuries in deeds of courage
and defiance. In my veins yet pulses the same hot blood that
sent Lapulapu to battle against the alien foe, that drove Diego
Silang and Dagohoy into rebellion against the foreign
oppressor.
That seed is immortal. It is the self-same seed that flowered
in the heart of Jose Rizal that morning in Bagumbayan when
a volley of shots put an end to all that was mortal of him and
made his spirit deathless forever; the same that flowered in
the hearts of Bonifacio in Balintawak, of Gregorio del Pilar
at Tirad Pass, of Antonio Luna at Calumpit, that bloomed in
flowers of frustration in the sad heart of Emilio Aguinaldo at
Palanan, and yet burst forth royally again in the proud heart
of Manuel L. Quezon when he stood at last on the threshold
of ancient Malacanang Palace, in the symbolic act of
possession and racial vindication. The seed I bear within me
is an immortal seed.
15
It is the mark of my manhood, the symbol of my dignity as
a human being. Like the seeds that were once buried in the
tomb of Tutankhamen many thousands of years ago, it shall
grow and flower and bear fruit again. It is the insigne of my
race, and my generation is but a stage in the unending
search of my people for freedom and happiness.
I am a Filipino, child of the marriage of the East and the
West. The East, with its languor and mysticism, its passivity
and endurance, was my mother, and my sire was the West
that came thundering across the seas with the Cross and
Sword and the Machine. I am of the East, an eager
participant in its struggles for liberation from the imperialist
yoke. But I know also that the East must awake from its
centuried sleep, shake off the lethargy that has bound its
limbs, and start moving where destiny awaits.
For I, too, am of the West, and the vigorous peoples of the
West have destroyed forever the peace and quiet that once
were ours. I can no longer live, a being apart from those
whose world now trembles to the roar of bomb and cannon
shot. For no man and no nation is an island, but a part of the
main, and there is no longer any East and West – only
individuals and nations making those momentous choices
that are the hinges upon which history revolves. At the
vanguard of progress in this part of the world I stand – a
forlorn figure in the eyes of some, but not one defeated and
lost. For through the thick, interlacing branches of habit and
custom above me I have seen the light of the sun, and I
know that it is good. I have seen the light of justice and
equality and freedom, my heart has been lifted by the vision
of democracy, and I shall not rest until my land and my
people shall have been blessed by these, beyond the power
of any man or nation to subvert or destroy.
I am a Filipino, and this is my inheritance. What pledge
shall I give that I may prove worthy of my inheritance? I
shall give the pledge that has come ringing down the
corridors of the centuries, and it shall be compounded of the
joyous cries of my Malayan forebears when first they saw
the contours of this land loom before their eyes, of the battle
cries that have resounded in every field of combat from
Mactan to Tirad Pass, of the voices of my people when they
sing:
“I am a Filipino born to freedom, and I shall not rest until
freedom shall have been added unto my inheritance—for
myself and my children and my children’s children—
forever.”