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Lesson Plan on Solve Right Triangles
Business math (Western Visayas College of Science and Technology)
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A Detailed Lesson Plan in Mathematics 9
I. Objectives
At the end of the lesson, eighty percent (80%) of the students will be able to:
1. integrate the Pythagorean theorem, leg angles, and trigonometric ratios in
solving a right triangle;
2. devise a strategy on solving a right triangle based on the given parts; and
3. solve for the missing parts of a right triangle.
II. Subject Matter
a. Topic:
Solving a right triangle by finding the measure of its parts.
b. References:
1. Mathematics Learner’s Material 9, pp. 437 – 440.
2. Art of Problem Solving (aops.com), Trigonometry in Right Triangles –
Article.
3. Brilliant.org (brilliant.org), Right Triangles and Trigonometric Ratios –
Articles.
c. Materials:
DLP, Laptop, Catolina, Calculator, and Manila Paper.
d. Value Focus:
To appreciate trigonometry and its application in searching the missing parts
of the triangle.
e. Skills:
Calculator proficiency, and critical thinking
III. Developmental Activities:
Teacher’s Activity
A. Preliminary Activities
1. Drills
1. Drills
Student’s Activity
a. Prayer
a. Prayer
A student leads the prayer.
b. Greetings
b. Greetings
Good morning class!
Good morning sir Jimenez
What makes your day complete?
(Different Student Responses)
2. Motivation
The Howlers
2. Motivation
The Howlers
Speaking of complete, have you ever
watched the movie <The 300=?
Yes sir!
(Different student responses)
What do you think that makes that movie
complete?
(The teacher will show a video clip in the
movie <The 300=.)
In the clip, what to do think are they doing?
(Different student responses)
Shouting a yell sir!
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The same with the movie clip, you will be
given a chance to create your own short
yell or howl. However, you have to do it
fast because if you are not ready within
three minutes, your group will be
disqualified for the upcoming game.
(Each group creates their own yell in three
minutes.)
Time is up, let me hear the howl of:
Group 3!
Group 2!
Group 1!
(Group 1 presents their howl.)
(Group 2 presents their howl.)
(Group 3 presents their howl.)
Here’s the game, whenever I say your
group, you will say your howl, okay? If you
have successfully complete the exchange of
howling with the other group without a
mistake, you will receive a reward!
Yes sir!
(The teacher will say the group their
names until two groups have been
eliminated.)
(Students participate with the game.)
Congratulations Group ____!
(Group ____ will say their howl.)
Here is the twist! In this whole period,
every time I’ll call the name of your group,
you will shout your howl. The best
performing group with receive a reward.
3. Review
3. Review
Alright! Before we formally start our
discussion, let’s try recall some of the basic
concepts in our right triangle.
A. Pythagorean Theorem
A. Pythagorean Theorem
What can you see on the screen?
(Projecting a figurative description of the
Pythagorean Theorem.)
(Different student responses.)
Is there any theorem you can associate this
image with?
Pythagorean Theorem sir!
What is Pythagorean Theorem?
The sum of the square of the legs of the
right triangle is equal to the square of the
hypotenuse.
For example, we have a triangle whose legs
are 4 and 3, what is the length of the
hypotenuse?
5 sir!
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B. Sum of the Interior Angles of a Triangle
B. Sum of the Interior Angles of a Triangle
In a triangle, what is the sum of all the
interior angles of a triangle?
For example, we have a right triangle with
one acute angle measuring 34°, find the
other angle?
180 degrees sir!
In any right triangle, what angle is common
to all of them?
90° sir!
If we deduct that 90° angle in the sum of
the three angles, what measure remains?
90°
And this remaining 90° represents _______.
The sum of the other two angles or the leg
angles.
C. Six Trigonometric Ratio
C. Six Trigonometric Ratio
Speaking of angles and sides, we have
mathematical concept that could relate the
two: _________________.
The six trigonometric ratio.
180° 2 90° 2 34°
56°
Who can recall our acronym for the six
trigonometric ratio?
SOH-CAH-TOA-CHO-SHA-CAO
Which short for ________________.
sin �㔃 = /�㕦āĀþăÿÿýă
cos �㔃 = /�㕦āĀþăÿÿýă
sec �㔃 =
cot �㔃 =
But today, we will only focus with sin �㔃,
cos �㔃, and tan �㔃.
4. Presentation of the Lesson
Now let us try to connect these three major
concepts by solving the right triangle.
What do we mean by solving a right
triangle?
Very good! With that, we shall:
1. integrate the Pythagorean theorem, leg
angles, and trigonometric ratios in solving
a right triangle;
2. devise a strategy on solving a right
triangle based on the given parts; and
3. solve for the missing parts of a right
triangle.
We have Pythagorean theorem, sum of
interior angles, and trigonometric ratio.
Now we have triangle ABC with right angle
at C.
ĀāāĀýÿþă
ÿĂĀÿāăÿþ
ĀāāĀýÿþă
tan �㔃 = ÿĂĀÿāăÿþ
/�㕦āĀþăÿÿýă
csc �㔃 = ĀāāāĀýÿþă
/�㕦āĀþăÿÿýă
ÿĂĀÿāăÿþ
4. Presentation of the Lesson
ÿĂĀÿāăÿþ
ĀāāĀýÿþă
Completing the sides and angles of a right
triangle.
(Students shall set expectation for the
class.)
ý
Ā
ÿ
ā
ÿ
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What theorem or concept can we relate
with:
1. ÿ, Ā, ā
2. ∠ý and ∠þ
3. ∠ý, ÿ and ā
4. ∠ý, Ā and ā
5. ∠ý, ÿ and Ā
6. ∠þ, ÿ and ā
7. ∠þ, Ā and ā
8. ∠þ, ÿ and Ā
These relationships shall lead us to our
activity!
1. Pythagorean Theorem
2. ∠ý + ∠þ = 90°
ÿ
3. sin ý = ā
Ā
4. cos ý = ā
ÿ
5. tan ý = Ā
Ā
6. sin þ = ā
ÿ
7. cos þ = ā
Ā
8. tan þ = ÿ
B. Lesson Proper
1. Activity
1. Activity
The Five Different Cases on Solving the
Right Triangle
The Five Different Cases on Solving the
Right Triangle
Group 1: Given the leg and the hypotenuse.
Group 2: Given an angle and the
hypotenuse.
Group 3: Given the leg and an acute angle
(opposite).
Group 4: Given the leg and an acute angle
(adjacent).
Group 5: Given the two legs.
Group 1: Given the leg and the hypotenuse.
Group 2: Given an angle and the
hypotenuse.
Group 3: Given the leg and an acute angle
(opposite).
Group 4: Given the leg and an acute angle
(adjacent).
Group 5: Given the two legs.
Setting of Standards/Instruction
Setting of Standards/Instruction
a. The class will be divided into five (5)
groups.
b. Each group will be given a marker and a
manila paper outlined for the activity.
c. On the manila paper, you will fill up the
necessary columns to complete the table.
d. After five (5) minutes, the group will
post their answers.
e. The presentation will be of much help in
your activity.
2. Analysis
Figure (true to all groups)
ÿ
e. Students shall look at the presentation
while answering.
2. Analysis
Figure (true to all groups)
ý
Ā
a. The class will be divided into five (5)
groups depending on their seating
arrangement.
b. Each group will receive a marker, and a
manila paper outlined for the activity.
c. Students fill up the columns required to
complete the table.
d. Students utilize the given time.
ý
ā
ÿ
Ā
þ
ÿ
ā
ÿ
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Let us now check your activity.
Given ÿ and ā, find
Ā
Given ∠ý and ā, find
ÿ
What equation
can I make?
What equation
can I make?
Given ∠ý, find ∠þ
For Group 5:
Case 5: Leg-Leg
Trigonometric
ratio
sin ý =
Given ∠ý and ā, find
Ā
Trigonometric
ratio
cos ý =
Given ∠ý, find ∠þ
Leg angles
Step 1
What equation
can I make?
What concept
should I use?
ÿ
ā
Ā
ā
∠ý + ∠þ = 90°
What concept
should I use?
What equation
can I make?
Trigonometric
ratio
tan ý =
Given ∠ý and ÿ,
find ā
Trigonometric
ratio
sin ý =
Given ∠ý, find ∠þ
Leg angles
∠ý + ∠þ = 90°
Condition
Given ∠ý and ÿ,
find Ā
ÿ
Ā
ÿ
ā
For Group 4:
Case 4: Angle-Leg (adjacent)
What equation
can I make?
What concept
should I use?
What equation
can I make?
Trigonometric
ratio
tan ý =
Given ∠ý and Ā,
find ā
Trigonometric
ratio
cos ý =
Given ∠ý, find ∠þ
Leg angles
∠ý + ∠þ = 90°
G4
Condition
Given ∠ý and Ā,
find ÿ
Step 2
Given ∠ý and Ā,
find ā
What concept
should I use?
G3
Step 3
Step 3
Step 2
Step 1
For Group 4:
Case 4: Angle-Leg (adjacent)
Given ∠ý and Ā,
find ÿ
ÿ2 + Ā 2 = ā 2
Given ∠ý and ā, find
ÿ
Step 2
What concept
should I use?
Given ∠ý, find ∠þ
Condition
Pythagorean
Theorem
Given ÿ and ā, find
Ā
ÿ
ā
For Group 3:
Case 3: Angle-Leg (opposite)
Given ∠ý and ÿ,
find ā
G4
∠ý + ∠þ = 90°
Condition
Step 3
Step 3
Step 2
Step 1
For Group 3:
Case 3: Angle-Leg (opposite)
Given ∠ý and ÿ,
find Ā
Leg angles
Using the ∠ý in step
1, find ∠þ
G2
Step 3
Step 3
What concept
should I use?
Given ∠ý, find ∠þ
Condition
sin ý =
For Group 2:
Case 2: Angle-Hyp
Given ∠ý and ā, find
Ā
G3
Trigonometric
Ratio
Given ÿ and ā, find
∠ý
Step 2
Condition
Step 2
G2
Step 1
For Group 2:
Case 2: Angle-Hyp
What equation
can I make?
Condition
Step 3
Using the ∠ý in step
1, find ∠þ
What concept
should I use?
G1
Step 2
What equation
can I make?
Step 1
Step 3
What concept
should I use?
Step 1
Given ÿ and ā, find
∠ý
For Group 1:
Case 1: Leg-Hyp
Step 1
Condition
Step 2
G1
Step 1
For Group 1:
Case 1: Leg-Hyp
For Group 5:
Case 5: Leg-Leg
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ÿ
Ā
Ā
ā
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What concept
should I use?
What equation
can I make?
G5
Step 1
Condition
Step 2
Given ÿ and Ā, find
∠ý
Using ∠ý in step 2,
find ∠þ
Step 3
Step 3
Step 2
Step 1
G5
Given ÿ and Ā, find
ā
Condition
Given ÿ and Ā, find
∠ý
Using ∠ý in step 2,
find ∠þ
Given ÿ and Ā, find
ā
What concept
should I use?
What equation
can I make?
Trigonometric
Ratio
tan ý =
Leg Angles
∠ý + ∠þ = 90°
Pythagorean
Theorem
ÿ2 + Ā 2 = ā 2
ÿ
Ā
3. Abstraction
3. Abstraction
Now, observe your answers on the activity.
What can you say about your entries on
that table?
Possible answers to be tackled:
1. Some tables have the same pattern on
entry in the concepts.
2. Tables with same concept entries have
the same first-word title.
3. When the table starts with Leg, its
concepts are in the pattern:
Trigonometric Ratio → Leg Angles→
Pythagorean
4. When the table starts with Angle, its
concepts are in the pattern:
Trigonometric Ratio → Trigonometric
Ratio → Leg angles
Is the step by step pattern on the table the
only to solve the right triangle? Why or
Why not?
No sir! It depends on your mastery to use
the theorems and concepts.
Furthermore, observe table 1. What if the
given is Ā and ā, is the step-by-step pattern
still applicable?
Yes sir! Since we are the ones labelling the
triangle, we can interchange their labels.
So what will happen to the formula?
All of the ý and ÿ will become þ and Ā, and
vice versa.
Is this method true to all of the tables?
Yes sir!
Very good! How about let’s apply this
concept on the go!
4. Application
1. Triangle BCA is right-angled at C. If ā =
23 and Ā = 17, find ∠ý, ∠þ and ÿ. Express
your answers up to two decimal places.
þ
ÿ
ÿ
ā = 23
Ā = 17
ý
4. Application
Case: Leg-Hyp
Step 1: Trigonometric Ratio to solve for ∠þ
Ā
sin þ =
ā
17
sin þ =
23
sin þ = 0.7391
Using the calculator
∠þ = 47.65°
Step 2: Since the sum of legs angles is 90°
∠ý + ∠þ = 90°
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∠ý + 47.65° = 90°
∠ý = 90° 2 47.65°
∠ý = 42.35°
2. Triangle ACB is right-angled at C. If
ÿ = 18.5 and Ā = 14.2, find ā, ∠ý and ∠þ.
ý
ā
Ā = 14.2
ÿ
þ
ÿ = 18.5
Step 3: Using Pythagorean Theorem
to get ÿ
ÿ2 + Ā 2 = ā 2
ÿ2 + 172 = 232
ÿ2 + 289 = 529
ÿ2 = 240
√ÿ2 = √240
ÿ = 15.49
Case: Leg-Leg
Step 1: Trigonometric Ratio to solve for ∠ý
ÿ
tan ý =
Ā
18.5
tan ý =
14.2
tan ý = 1.3028
Using the calculator
∠ý = 52°
Step 2: Since the sum of legs angles is 90°
∠ý + ∠þ = 90°
52° + ∠þ = 90°
∠þ = 90° 2 52°
∠þ = 38°
Step 3: Using Pythagorean Theorem
to get ā
ÿ2 + Ā 2 = ā 2
18.52 + 14.22 = ā 2
342.25 + 201.64 = ā 2
543.89 = ā 2
√ā 2 = √543.89
ā = 23.32
3. Triangle BCA is right-angled at C if
ā = 27 and ∠ý = 58°, find ∠þ, Ā, and ÿ.
þ
ā = 27
ÿ
ÿ
Ā
58°
ý
Case: Angle-Hyp
Step 1: Trigonometric Ratio to solve for ÿ
ÿ
sin 58° =
ā
ÿ
sin 58° =
27
ÿ = 27 sin 58°
Using the calculator
sin 58° = 0.8480
Hence,
ÿ = 27 (0.8480)
ÿ = 22.9
Step 2: Trigonometric Ratio to solve for Ā
Ā
cos 58° =
ā
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Ā
27
Ā = 27 cos 58°
Using the calculator
cos 58° = 0.5299
Hence,
Ā = 27 (0.5299)
Ā = 14.31
cos 58° =
4. Triangle ACB is right-angled at C. If
∠ý = 63° and ÿ = 11, find ∠þ, Ā and ā.
ý
Ā
63°
ÿ
ā
ÿ = 11
þ
Step 3: The sum of the legs angles is 90
degrees, so
∠ý + ∠þ = 90°
58° + ∠þ = 90°
∠þ = 90° 2 58°
∠þ = 32°
Case: Angle – Leg (Opposite)
Step 1: Trigonometric Ratio to solve for Ā
ÿ
tan 63° =
Ā
11
tan 63° =
Ā
11
Ā=
tan 63°
Using the calculator
tan 63° = 1.9626
Hence,
11
Ā=
1.9626
Ā = 5.60
Step 2: Trigonometric Ratio to solve for ÿ
ÿ
sin 63° =
ā
11
sin 63° =
ā
11 = ā sin 63°
11
ā=
sin 63°
Using the calculator
sin 63° = 0.8910
Hence,
11
ā=
0.8910
ā = 12.35
I think you are ready for a quiz!
Step 3: The sum of the legs angles is 90
degrees, so
∠ý + ∠þ = 90°
63° + ∠þ = 90°
∠þ = 90° 2 63°
∠þ = 27°
IV. Evaluation:
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In one-half crosswise, answer the following problems. Express your answers up to two
decimal places and evaluate trigonometric ratios by four decimal places.
1. Triangle BCA is right-angled at C. If ā = 23 and Ā = 17,
þ
find ∠ý, ∠þ and ÿ.
ā = 23
ÿ
ÿ
þ
Ā = 17
ā
ÿ
ÿ
ý
63°
Ā = 11
2. Triangle ACB is right-angled at C. If ∠ý = 63° and Ā = 11, find ∠þ, ÿ and ā.
ý
V. Assignment
Use the given figure to solve the remaining parts of right triangle ACB.
1. Ā = 17 and ā = 23
2. ā = 16 and ÿ = 7
3. ý = 15° and ā = 37
4. þ = 30° and Ā = 11
5. þ = 18° and ÿ = 18
þ
ā
ÿ
ÿ
ý
Ā
VI. Remediation
þ
ÿ
ÿ
ā
Ā
Use the figure below, write the expression that gives the required unknown.
1. If ý = 15° and ā = 37, find ÿ
2. If ý = 49° and ÿ = 10, find ā
3. If ÿ = 13 and þ = 16°, find ā
4. If ā = 16 and ÿ = 7, find Ā
5. If ÿ = 7 and Ā = 12, find ý
ý
VII. Enrichment
Sketch a figure and solve each right triangle ABC with right angle at ÿ, given that:
1. ÿ = 15.8, Ā = 21
2. ÿ = 7, Ā = 12
3. ÿ = 2, Ā = 7
4. ÿ = 3, Ā = √5
5. ÿ = 250, Ā = 250
Prepared by:
Name of Student
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