Mathematics
9
Quarter 3
Self-Learning Module 13
45o-45o-90o
Right Triangle Theorem
Mathematics Grade 9
Quarter 3 – Self-Learning Module 13: 𝟒𝟓𝐎-𝟒𝟓𝐎-𝟗𝟎𝐎 Right Triangle Theorem
First Edition, 2020
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Mathematics
9
Quarter 3
Self-Learning Module 13
𝟒𝟓𝐎 -𝟒𝟓𝐎 -𝟗𝟎𝐎 Right Triangle Theorem
Introductory Message
For the Facilitator:
Welcome to the Mathematics Grade 9 Self-Learning Module on 45O -45O -90O
Right Triangle Theorem!
This Self-Learning Module was collaboratively designed, developed and
reviewed by educators from the Schools Division Office of Pasig City headed by its
Officer-in-Charge Schools Division Superintendent, Ma. Evalou Concepcion A.
Agustin, in partnership with the City Government of Pasig through its mayor,
Honorable Victor Ma. Regis N. Sotto. The writers utilized the standards set by the K
to 12 Curriculum using the Most Essential Learning Competencies (MELC) in
developing this instructional resource.
This learning material hopes to engage the learners in guided and independent
learning activities at their own pace and time. Further, this also aims to help learners
acquire the needed 21st century skills especially the 5 Cs, namely: Communication,
Collaboration, Creativity, Critical Thinking, and Character while taking into
consideration their needs and circumstances.
In addition to the material in the main text, you will also see this box in the
body of the self-learning module:
Notes to the Teacher
This contains helpful tips or strategies that
will help you in guiding the learners.
As a facilitator you are expected to orient the learners on how to use this selflearning module. You also need to keep track of the learners' progress while allowing
them to manage their own learning. Moreover, you are expected to encourage and
assist the learners as they do the tasks included in the self-learning module.
For the Learner:
Welcome to the Mathematics Grade 9 Self-Learning Module on 45O -45O -90O
Right Triangle Theorem!
This self-learning module was designed to provide you with fun and
meaningful opportunities for guided and independent learning at your own pace and
time. You will be enabled to process the contents of the learning material while being
an active learner.
This self-learning module has the following parts and corresponding icons:
Expectations - This points to the set of knowledge and skills
that you will learn after completing the module.
Pretest - This measures your prior knowledge about the lesson
at hand.
Recap - This part of the module provides a review of concepts
and skills that you already know about a previous lesson.
Lesson - This section discusses the topic in the self-learning
module.
Activities - This is a set of activities that you need to perform.
Wrap-Up - This section summarizes
application of the lesson.
the
concepts
and
Valuing - This part integrates a desirable moral value in the
lesson.
Posttest - This measures how much you have learned from the
entire module.
EXPECTATIONS
1. State and explain 45O -45O -90O Right Triangle Theorem.
2. Solve problems applying the 45O -45O -90O Right Triangle Theorem.
PRETEST
Directions: Read each of the following questions carefully. Write only the letter of
the correct answer on your paper.
1. Which of the following is the other term for 45O -45O -90O Right Triangle?
A. Isosceles Right Triangle
C. Scalene Right Triangle
B. Isosceles Triangle
D. Scalene Triangle
O
O
O
2. In a 45 -45 -90 Right Triangle, the _________ is √2 times the length of any of the
legs.
A. Altitude
C. Longer Leg
B. Hypotenuse
D. Shorter Leg
3. Refer to the figure at the right, solve the value of x.
A. 1 cm
C. 2√2 cm
B. √2 cm
D. 4 cm
4. Which of the following must be the length of the hypotenuse whose leg measures
5 cm each?
A. √5 cm
C. 5√2 cm
B. 2√5 cm
D. 10 cm
5. What is the area of the square whose diagonal is 6 cm long?
A. 12 cm2
C. 36 cm2
2
B. 18 cm
D. 72 cm2
RECAP
LET’S FIND IT!
Direction: Find the length of the following segments.
1. r = _________
4. x = _________
2. s = _________
5. y = _________
3. t = _________
LESSON
Square is a quadrilateral that has four equal sides and four right angles. When
a square is cut into halves diagonally, two right triangles are formed. These right
triangles are called Isosceles Right Triangles or the 𝟒𝟓𝐎-𝟒𝟓𝐎-𝟗𝟎𝐎 right triangles
because they have two equal sides and a right angle. On the other hand, if an
isosceles triangles is cut into halves by perpendicular bisector, another pair of right
triangle are formed where the interior angles are 𝟑𝟎𝐎-𝟔𝟎𝐎-𝟗𝟎𝐎.
The triangles being referred to above are generally called SPECIAL RIGHT
TRIANGLES because they have some characteristics or properties that are very
useful in solving problems concerning right triangles.
THE 𝟒𝟓𝐎 -𝟒𝟓𝐎-𝟗𝟎𝐎 RIGHT TRIANGLE THEOREM
In an isosceles right triangle, the hypotenuse is √2 times as long as each of
the legs.
Given: ∆STQ is an isosceles right triangle with ST = TQ = x.
SQ = t and m∠T = 90
Prove: t = √2x
Statements
1. = x 2 + x 2
2. t 2 = 2x 2
Reasons
The Pythagorean Theorem.
Addition of Similar Terms.
Extraction of the Square Roots on both sides.
t2
3. t = √2 x
Examples:
A. Complete the table below.
1.
2.
3.
LEGS (x, z)
8cm
HYPOTENUSE (y)
15 cm
7√2 cm
4.
5.
X
√3
2
√5
4
cm
cm
45O
LEG
(z)
45O
Y
LEG (x)
Z
SOLUTIONS:
1. Solving for hypotenuse.
We are going to multiply the length of the legs by √2.
(8)(√2) = 8√2 cm
Therefore the length of the hypotenuse is 8√2 cm.
2. Solving for the legs.
We are going to divide the length of the hypotenuse by √2.
15
Divide 15 by √2.
√2
15
√2
√2
√2
∙
15√2
=
15√2
Rationalize the denominator.
√4
cm
2
Simplify.
Therefore the length of the legs is
15√2
2
cm.
3. Solving for hypotenuse.
We are going to multiply the length of the legs by √2.
(7√2)) (√2)
7√4
Multiply
7(2)
Simplify
14 cm
Therefore the length of the hypotenuse is 14 cm.
4. Solving for the legs.
We are going to divide the length of the hypotenuse by √2.
√3
2
Divide
√2
√3
2
1
∙
∙
√6
2(2)
=
by √2.
Get the reciprocal of the denominator
√2
√3
2√2
√3
5
then multiply
√2
√2
=
√6
4
√6
2√4
cm
Therefore the length of the legs is
Rationalize the denominator
Simplify
√6
4
cm.
5. Solving for hypotenuse.
We are going to multiply the length of the legs by √2.
(
√5
4
) (√2) =
√10
4
cm
Therefore the length of the hypotenuse is
√10
4
cm.
B. Find the area of a square whose diagonal
measures 3 ft. long.
Solution:
First we solve for the length of the sides of the square.
3
√2
∙
√2
√2
3 ft.
=
=
3√2
√4
3√2
2
ft.
Therefore each side measures
3√2
2
ft.
We can now solve for the area of the square using the length of the side.
Area of the Square = side2
Area of the Square = (
3√2
2
Area of the Square =
3√2
2
Area of the Square =
9√4
4
Area of the Square =
18
4
∙
or
2
𝑓𝑡. )
3√2
2
9
2
Therefore the area of the square is
ft2
9
2
ft2.
ACTIVITIES
ACTIVITY 1: LET’S PRACTICE!
Directions: Use the figure at the right to complete the table below. Write your
answers on the space provided.
LEGS (g, q)
1.
2.
3
16 m
4.
5√7
4
HYPOTENUSE (c)
5√2 m
1
2
m
m
5.
7√11
5
m
ACTIVITY 2: KEEP PRACTICING!
Directions: Find the length of the missing sides of the following figures. Write your
answers on the space provided.
1.
2.
3.
ACTIVITY 3: TEST YOURSELF!
Directions: Answer the following problems correctly. Show your complete solution.
1. The length of the hypotenuse of a 45O -45O -90O triangle is 10√3 m. Find the
length of the legs. (1pt.)
2. If the area of the square is 121 cm 2, find the length of the sides and diagonal
of the square. (2pts.)
3. Find the area and the sides of the square whose diagonal is
7√3
2
cm. (2pts.)
WRAP–UP
Remember that…
There are two types of special right triangles.
•
•
45O -45O -90O Right Triangle
30O -60O -90O Right Triangle
The 𝟒𝟓𝐎-𝟒𝟓𝐎-𝟗𝟎𝐎 RIGHT TRIANGLE THEOREM states that “In an isosceles right
triangle, the hypotenuse is √2 times as long as each of the legs”.
45O -45O -90O Right Triangle is also known as Isosceles Right Triangle.
VALUING
REFLECTION: (Journal Writing)
Right-angled triangles are used alongside trigonometry to solve real-world
distance problems, such as the distance of a ladder of a known length can go up
against the wall. If the angle the ladder makes with the ground is also known. In life,
success is not a matter of good fortune or an accident of birth. It’s a matter of
decision, commitment, planning, preparation, and execution. If you want to succeed
in life, would you like to follow the right ladder to success? Do you have a dream to
follow? Are you willing to take the risk along your way of struggle?
POSTTEST
Directions: Read each of the following questions carefully. Write only the letter of
the correct answer on your paper.
1. In a 45O -45O -90O right triangle, the hypotenuse is √2 times the length of any of
the ________.
A. Altitude
C. Hypotenuse
B. Base
D. Legs
2. In an Isosceles Right Triangle, Which of the following will be the length of the
hypotenuse if its legs measures 2√2 m?
A. 4 m
C. 16 m
B. 8 m
D. 32 m
MODULE 13
PRETEST
1. A
2. B
RECAP
3. B
1. 9 m
2. 3√13 m
3. 2√13 m
ACTIVITY 1: LET’S PRACTICE!
1. 16√2 m
2. 5 m
3.
√2
4
m
4. C
4. 3√10 m
4.
5√14
4
m
5. B
5. 9√10 m
5.
7√22
10
m
ACTIVITY 2: KEEP PRACTICING!
1. x = √26 ft. ; y = √13 ft.
2. x =
6√10
5
ft. ; y =
6√5
ft.
5
5. x =
9√3
4
ft.
ACTIVITY 3: TEST YOURSELF!
1. 5√6 m
2. 𝑠𝑖𝑑𝑒𝑠 = 11 cm; diagonal = 11√2 cm
3. 𝑠𝑖𝑑𝑒𝑠 =
7√6
4
cm, Area =
POSTTEST
1. D
2. A
147
𝑐𝑚2
8
3. B
4. C
5. B
KEY TO CORRECTION
5. What is the area of the square whose diagonal is 10 cm long?
A. 10 cm2
C. 250 cm2
B. 50 cm2
D. 625 cm2
A. 5 ft.
B. 5√2 ft.
C. 10√2 ft.
D. 20 ft.
4. Which of the following will be the length of the diagonal of a square whose side
measures 10 ft. long?
B.
A.
√3
m
4
√6
m
4
D.
C.
√3
2
√6
2
m
m
3. Given the figure at the right, what must be the value of x?
References
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2004.
Bernabe, Julieta, Dilao, Soledad Jose, and Cecile De Leon. Geometry III. Quezon City:
JTW Corporation, 2002.
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Vera, Richard F., Garcia, Gilda T., Javier, Sonia E., Lazaro, Roselle A.,
Mesterio, Bernadeth J., and Rommel Hero A. Saladino. Mathematics 9
Learner’s Module. Department of Education Philippines., (1st Edition 2015).
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Cruz. Geometry III. Quezon City: Bookman, Inc., 2003.
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M., & N. Villas. Grade 9 Mathematics Patterns and Practicalities. Chino Roces
Ave., Makati: Salessiana Books, Don Bosco Press, Inc., 2013.
Orines, Fernando, Mercado, Jesus, and Josephine Suzara. Next Century Mathematics
Geometry. Quezon City: Phoenix Publishing House, 2008.
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2007.
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