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Republic of the Philippines
Department of Education Regional Office IX, Zamboanga Peninsula
Mathematics
Zest for Progress
Zeal of Partnership
Quarter 3 - Module 7: Triangle Similarity Theorems Application
and Proof of Pythagorean Theorem
9
Name of Learner: ___________________________
Grade & Section: ___________________________
Name of School: ___________________________
1
Good Day Scout!
In the previous module, you learned about triangle similarity theorems and you have proven each of
them. In this module, you will learn to apply those theorems to show that the given triangles are similar.
You will also learn how to prove the Pythagorean Theorem.
WHAT I NEED TO KNOW
WHAT I KNOW
Find out how much you already know about this lesson. Encircle the letter of the correct
answer. Take note of the items that you were not able to answer correctly and find out the right answer as you go through this module.
1. If a segment bisects an angle of a triangle, then it divides the opposite side into segments
proportional to the other two sides. What do you call that segment?
a. height
b. altitude
c. angle-bisector
d. shorter leg
2. If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are
similar to the original triangle and to each other. Identify the altitude in the given right triangle.
a. ME
b. ER
c. EY
d. MR
3. Which method can be used to show that the two triangles at the right are similar?
a. AA Similarity Theorem
b. SAS Similarity Theorem
c. SSS Similarity Theorem
d. Triangle Proportionality Theorem
Module 7
APPLYING TRIANGLE SIMILARITY THEOREMS AND PROVING THE PYTHAGOREAN THEOREM
LEARNING COMPETENCY
In this module, you will be able to:
apply the triangle similarity theorems to show that given triangles are similar (M9GE-IIIi-1)
prove the Pythagorean Theorem (M9GE-IIIi-2)
?
15
6
20 8
M Y
R
E
2
4.In ABCD EFGH. Which similarity theorem that makes BCD FGH?
a. AA Similarity Theorem
b. SAS Similarity Theorem
c. SSS Similarity Theorem
d. Triangle Proportionality Theorem
5. In the diagram below, KLM NPQ. What is the length of KL?
a. 6 m
b. 12 m
c. 24 m
d. 26 m
6. ABC DEF. What is the perimeter of DEF?.
a. 9
b. 7
c. 8
d. 6
7. Which are NOT sides of a right triangle?
a. 3,4,5 b. 9,24,25 c. 9,12,15 d. 8,15,17
8. in the figure at the right, What is the value of x?
a. 13
b. 12
c. 11
d. 10
9. Given the illustration below, find the missing side length.
a. 10
b. 9
c. 8
d. 7
10. For any right triangle, the sum of the squares of the lengths of the legs equals the square of
the length of the hypotenuse. Which of the following illustration below is a proof of Pythagorean
Theorem?
Congratulations Scout!
I am looking forward for more progress and as for now, here is your first Merit Badge
Keep going and receive more badges.
15
12
?
3
WHAT’S IN
Do you know that Engineers used similar triangles when
designing buildings?
Example: Pyramid Building in San Diego, California?
There are several ways to prove certain triangles are
similar. SAS, SSS, AA Similarity Theorems along with the
Right Triangle and Special Right Triangle Theorems were
used to prove triangle similarity.
Angle-Angle (AA) Similarity Hypothesis Conclusion
If two angles of one triangle are
congruent to two angles of another ABC ~ DEF
triangle, then the triangles are similar.
Side-Side-Side (SSS) Similarity Hypothesis Conclusion
If the three sides of one triangle
are proportional to the three corresponding ABC ~ DEF
sides of another triangle, then the triangles
are similar.
Side-Angle-Side (SAS) Similarity Hypothesis Conclusion
If two sides of one triangle are
proportional to two sides of another triangle ABC ~ DEF
and their included angles are congruent,
then the triangles are similar.
Right Triangle Similarity Theorem Hypothesis Conclusion
If the altitude is drawn to the
hypotenuse of a right triangle, then the BAC ADC BDA
two triangles formed are similar to the
original triangle and to each other.
Special Right Triangle Theorems
30-60-90 Right Triangle Theorem
The shorter leg is ½ the hypotenuse h
or √
times the longer leg;
the longer leg l is √3 times the shorter leg (s); and
the hypotenuse h is twice the shorter leg.
45-45-90 Right Triangle Theorem
Each leg is √
times the hypotenuse; and
the hypotenuse is √2 times each leg l.
Jason DiSilva.,Pyramid Building in San Diego,California. www. flickr.com
B E
A
C F
D
A
C B
D
E F
C
B
A
F
E
D
A
C B D
so,
ℎ = √2𝑙 𝑙 =√
ℎ
so,
ℎ = 2𝑠 𝑙 = √3𝑠 𝑠 =√3
3𝑙 𝑜𝑟 𝑠 =
1
ℎ
4
Previously, you learned that the concepts above are used to prove triangle similarity. Now, answer the
activity below using the concepts you have learned in the previous lesson.
Grab this opportunity to receive another merit badge!
Activity 1: Am I Right or Wrong?
Directions: Read each statement carefully. Put a check () on the space provided before the
number if the statement is correct and (Ꭓ) if wrong.
____1. Two triangles are similar if two angles of one triangle are congruent to two angles of
another triangle.
____2. Two triangles are similar if the corresponding sides of two triangles are equal.
____3. Two triangles are similar if an angle of one triangle is congruent to an angle of another
triangle and
the corresponding sides including those angles are in proportion.
____4. If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed
are only similar to each other.
____5. In a 30-60-90 Right Triangle, the hypotenuse is twice the shorter leg.
You’re doing well, Scout!
You really are heedful! Here’s your second Merit Badge
Keep going
WHAT’S NEW
Activity 2: Obey My Command
Hey, Scout! Will you obey my commands? Let’s get started! Write your responses inside the box
provided and answer the questions below.
Leader says… Scout does…
Draw two triangles and name it ΔBSP and ΔLAW
where BS LA, SP AW and PB WL.
Label the sides of the triangles.
Let BS=16cm, SP=20cm, PB=12cm and WL=6cm
Now, find the value of LA and AW.
Follow-up Questions:
1. Are the two triangles similar?
Answer ___________________________________
____________________________________________
2. Which Triangle Similarity Theorem helps you prove your
answer?
Answer______________________________________________________________________
_____________________________________________________________________________
Wow! You have unlocked another Merit Badge
Keep it Up!
5
WHAT IS IT
Now, let’s learn how to apply the theorems and show that triangles are similar. You will also learn
how to prove the Pythagorean Theorem. Isn’t it exciting?
Application of SAS Similarity Theorem
Facts to consider!
1. We can say that triangles are similar by SAS Similarity Theorem if it involves two sides and
an angle between them.
2. Test the given. If it entails to a common ratio, then the triangles are similar.
3. Always consider the corresponding sides. The longest side of a triangle corresponds to the
longest side of the other triangle so are the other sides (see side colors).
Example 1: Are the two triangles similar? How do you know?
Therefore, ΔABC ΔXZY by SAS Similarity Theorem
Application of SSS Similarity Theorem
Facts to consider!
1. We can say that triangles are similar by SSS Similarity Theorem if it involves three sides.
2. Test the given. If it entails to a common ratio, then the triangles are similar.
3. Always consider the corresponding sides. The longest side of a triangle corresponds to the
longest side of the other triangle so are the other sides (see side colors)
Example 2: Are the two triangles similar? How do you know?
Therefore, ΔACB ΔEDF by SSS Similarity Theorem
Application of AA Similarity Theorem
Facts to consider!
1. We can say that triangles are similar by AA Similarity Theorem if interior angles of both
triangles are congruent.
2. Always remember that the sum of interior angles of a Triangle is equal to 1800.
B C
A
36
15
Z
Y
X
24
𝑍𝑋
𝐵𝐴=
10
15=
2
3
𝑍𝑌
𝐵𝐶=
24
36=
2
3
2
3 𝑖𝑠 𝑡ℎ𝑒 𝑐𝑜𝑚𝑚𝑜𝑛 𝑟𝑎𝑡𝑖𝑜
A F
B C E
D 21
28
14 10
20
15
𝐵𝐶
𝐹𝐷=
28
20=
7
5
𝐵𝐴
𝐹𝐸=
21
15=
7
5
𝐴𝐶
𝐸𝐷=
14
10=
7
5
7
5 𝑖𝑠 𝑡ℎ𝑒 𝑐𝑜𝑚𝑚𝑜𝑛 𝑟𝑎𝑡𝑖𝑜
6
Example 3: Are the two triangles similar? How do you know?
Find x.
E L ; F M ; G N Therefore, ΔEFG ~ ΔLMN by AA Similarity Theorem.
Application of Right Triangle Similarity Theorem
Example 4:
Application of Special Right Triangle Theorems
45-45-90 Right Triangle Theorem
Example 5:
E
G
F M
L
N
1020 1020
300
480 x
y
x= 1800-(1022+300) x= 1800-1320 x= 480
y= 1800-(1022+480) y= 1800-1500 y= 300
F
I A R 4cm 9cm
x
13cm
Breakdown:
F
R A F
I A
F
R I 4cm
9cm
13cm
x=6
x=6 𝑥
4=
9
𝑥
𝑥 𝑥 = 9 4 𝑥 = 36
𝑥 = √36 𝑥 = 6
B
A C
c a
b=3
a and b are the legs of the 45-45-90 triangle and they are equal.
c is the hypotenuse (longest side) and
it is equal to leg times √3.
so, a=3, b=3, and c=3√2
450
450
7
30-60-90 Right Triangle Theorem Example 6:
Pythagorean Theorem: Pythagoras Proof by Rearrangement How do we know that a theorem is true for every right triangle on a flat surface? Take four identical right triangles with side lengths a and b, and hypotenuse length c.
Arrange them so that their hypotenuse formed a square.
The area of this square now is c2, based from the area formula A=s2 where s, stands for the sides.
Now, rearrange the triangle forming two rectangles.
Here’s the Key! The total area of the two figure did not change and the areas of the triangles did not change
so, the empty space in Illustration 1 which is c2 must be equal to the empty space of illustration 2 which is a2+b2.
This proves that c2=a2+b2
Great Job, Scout! You’ve studied well. Here’s another Merit Badge for you
A C
B
Short
er
Leg
Longer Leg
a=2
b
c longer leg= shorter times √3
hypotenuse= 2 times shorter leg
so, b= 2√3 and c= 4
a
b
c a a a
b b b
c c c
c
c c
c
c2
b
b
a a
The area of the two squares are b2 and a2.
a2 b2
Illustration 1.
Illustration 2.
8
WHAT’S MORE Wow! You made it this far. Now, try to apply the theorems to determine whether the given triangles are similar or not.
Activity 3: Are We Similar? By What? Directions: Write TS on the space provided before the number if the triangles are similar and NS if they are not similar. Identify which theorem helps you determine your answer and write it on the box provided above the illustrations. (Angle-Angle (AA) Similarity, Side-Side-Side (SSS) Similarity, Side-Angle-Side (SAS) Similarity and Right Triangle Similarity Theorem) ____1. ____2. ____3. ____4. _____5.
Amazing! You did it well, Scout! Right now, you have earned your fifth Merit Badge
Keep going!
A
E
D
B
C
5
12 9
3
A
B
C
F
E D
9
18
15
12
6
10
A
B C E
D
F
10 21
15
25
35
20
810
600
400 810
A
B
C
D
E
F
20
32
y
24
x
21
L
E
G M
A
P
9
WHAT I HAVE LEARNED
Activity 4: What’s My Value? Directions: Analyze each illustration and solve the unknown values by applying the theorems you have learned. Write your answer on the box provided in each item. 1. 2. 3. Explain briefly. 1. How did you get the value of x? Answer____________________________________________________________________________ ____________________________________________________________________________________ 2. How did you solve for the shorter leg? The longer leg? Answer____________________________________________________________________________ ____________________________________________________________________________________ ____________________________________________________________________________________ 3. How did you find the value of the other leg and the hypotenuse? Answer____________________________________________________________________________ ____________________________________________________________________________________
Another Merit Badge Unlocked Congratulations! You are doing great, Scout!
M
A T
H
x=
8cm
26cm
600
300
G
S P
30ft.
450
450
B
P S
1.5m
10
WHAT I CAN DO Safety Application
To prevent a ladder from shifting, A safety experts recommended that the ratio of a: b be 4:1. How far from the base of the wall should you place the foot of a 10-foot ladder? Round to the nearest inch.
Let x be the distance in feet from the foot of the ladder to the base of the wall. Then 4x is the distance in feet from the top of the ladder to the base of the wall. a2 +b2 =c2 Pythagorean Theorem (4x)2+x2 = 102 Substitute 16x2+x2= 100 17x2=100 Multiply and combine like terms
=1
1 Divide both sides by 17
= √1
1 2ft. 5in. Find the positive square root and round it.
Now, it’s your turn!
Activity 5: Safety Application Problem: The safety rules for a playground state that the height of the slide and the distance from the base of the ladder to the front of the slide must be in a ratio 3:5. If a slide is about 16 feet long, what are the height of the slide and the distance from the base of the ladder to the front of the slide?
Hey! Scout, let safety be first in everything that you do! Be an advocate of Safety Environment
You deserve another Merit Badge
a
b
10ft.
12ft.
5x
3x
Solution:
11
ASSESSMENT
Directions: Analyze each question and choose the letter of the correct answer. Write the letter of your answer on the space provided before the number.
____1. Which triangle similarity theorem is used to prove that ΔGSP~ΔBSP? a. AA Similarity Theorem b. SAS Similarity Theorem c. SSS Similarity Theorem d. Right Triangle Similarity Theorem For items 2-3. Refer to the illustration below.
____2. Apply SAS Similarity Theorem. What is the ratio of the given corresponding sides?
a.
3 b.
5 c. 2 d.
1
____3. GT GA and GR GE. Solve for the value of x to prove that the two triangles are similar. a. 5cm b. 4cm c. 3cm d. 2cm
____4. Find the x and y, such that ΔSAY ~ Δ TWO.
a. 4 and 10 b. 4 and 12 c. 2 and 12 d. 2 and 10
G
S B
S
P
P 12
3
G
A
E
R
T
4cm
2cm
6cm
x
S
A
Y
T
O
W
9
18
4x-1
y
6
10
12
____5. Analyze the given illustration. Which of the following shows the accurate breakdown of triangles according to Right Triangle Similarity Theorem?
a.
b.
c.
d.
____6. Which value of x would make ΔERB ΔBRA similar?
a. 8 b. 9 c.10 d. 11
____7. A right triangle measures 8.5 cm, 14.72cm, and 17 cm each side. What kind of special right triangle it is? Why? a. A 45-45-90 Special Right Triangle because the shorter leg is half the hypotenuse.
b. A 30-60-90 Special Right Triangle because the longer leg is the product of √2 and the shorter leg. c. A 30-60-90 Special Right Triangle because the shorter leg is half the hypotenuse and the
longer leg is the product of the shorter leg and √3. d. It is not a special right triangle. ____8. ΔBIG is a right triangle with a shorter leg 8cm and a longer leg 17cm. What is the perimeter of ΔBIG? Round to the nearest hundredths. a. 42.79cm b. 43.79cm c. 47.392cm d. 43.29cm ____9. John climbed up on a Mango Tree. He was 5 meters above the ground when he decided to
climbed down but he suddenly stopped and asked his father for help. His father brought a 6m ladder, leaned it on the mango tree and help John to get down.
How far is the foot of the ladder from the base of the Mango Tree? a. 3.32m b. 4m c. 4.32m d. 3.92m
B
R A
E
B
A
E R
E
B R
B
A
B
B
B
A
A
A
E
E
E B
B
B
E
E
E
R
R
R A
A
A
B
B
B
R
R
R
13
____10. It is believed that surveyors in Ancient Egypt laid out right angles using a rope divided into twelve sections by eleven equally spaced knots.
How could the surveyors use this rope to make a right angle? a. The twelve sections with eleven equally spaced knots represents the perimeter of a
right triangle which sides are 3,4, and 5, a perfect example of Pythagorean Triple. b. The surveyors used knotted ropes to measure triangular areas considering the number
of knots. c. The rope divided into twelve sections by eleven equally spaced knots was laid out to a
triangular object with a right angle. d. The twelve knots were divided into three that makes a 4-units sided triangle.
Nothing’s really hard for you, Scout!
You’ve done it excellently. Here’s two Merit Badges for you
Additional Activity
Activity 6: Learning by Doing!
Directions: Look for a used rope or any substitute. Knot it with equal spaces between them. Use the knotted rope to form a right triangle. Consider each knot as one unit. List the sides of right triangles that perfectly forms a Pythagorean Triple. Give at least 5 and check. Write your answer on the space provided. Use extra sheet if necessary.
Example: Use the measure of the side of a triangle 3cm,4cm and 5cm. Since 5 is the longest side, let be the hypotenuse. 4 is the longer leg and 3 is the shorter leg.
SIDES
Checking Shorter Leg Longer Leg Hypotenuse
Sample:
3cm
4cm
5cm
Checking: c2 = a2 + b2 52 = 32 + 42 25 = 9 +16
25 = 25
3cm
4cm
5cm
14
Finally, you are done!
Tap yourself and say “I did it!” Add up your Merit Badges and convert it into number of hours of sleep.
Reward yourself by spending enough hours to sleep and rest
Mathematics 9 3RD QUARTER MODULE 7 Answer Key WHAT I KNOW (Pre-Test) 1. c 2. c 3. b 4. b 5. c 6. a 7. b 8. a 9. b 10. a WHAT’S IN (Activity 1) 1. / 2. x 3. / 4. x 5. /
15
Solution: a2 + b2 = c2
(3x)2 + (5x)2= 122
9x2 + 25x2 = 144
34x2 = 144
34𝑥2
34=
144
34
√𝑥 = √144
34
x=2.06 ft.
WHAT’S NEW (Activity 2) Scout does… 1. YES 2. SSS Similarity Theorem WHAT’S MORE (Activity 3) 1. NS- SAS SIMILARITY THEOREM 2. TS- SSS SIMILARITY THEOREM 3. NS – SSS SIMILARITY THEOREM 4. NS- AA SIMILARITY THEOREM 5. TS- AA SIMILARITY THEOREM WHAT I HAVE LEARNED (Activity 4) 1. x = 12
2. GS = 15ft., SP= 15√3 ft.
3. SP = 1.5m , BP 2.12m Explanation: 1. Applying the concept of Right Triangle Similarity Theorem.
2. Shorter leg is half the hypotenuse and the longer leg is shorter leg times √3.
3. Two legs are equal and the hypotenuse is equal to the leg times √2. WHAT I CAN DO (Activity 5) ASSESSMENT 1. a 2. a 3. c 4. b 5. a 6. c 7. c 8. b 9. a 10. a ADDITIONAL ACTIVITIES (Activity 6) Answer may vary. Please refer to the example and check Prepared by: JICELLE E. GEGRIMOS
B
S P W A
L
12 16
20 10
6 8
16
References Bryant, Merden L., Leonides E. Bulalayao, Melvin M. Callanta, Jerry D. Cruz, et al. Mathematics Teachers Guide 9. Pasig City: Department of Education, 2014 Bryant, Merden L., Leonides E. Bulalayao, Melvin M. Callanta, Jerry D. Cruz, et al. Learner’s Material Mathematics 9. Pasig City: Department of Education, 2014 Burger, Edward B. Ph.D., David J. Chard Ph.D., Earlene J. Hall Ed.D., Paul A. Kennedy Ph.D., et al., Holt Geometry: Houghton Mifflin Publishing Company,2011. TED-Ed’s. “How to prove Pythagorean Theorem. December 1, 2020” https://ed.ted.com CK-12 Foundation. “Triangle AA Similarity: Examples (Basic Geometry Concepts)”, “SAS Similarity Theorem: Lesson (Basic Geometry Concepts) and “SSS Similarity: Examples (Basic Geometry Concepts).” https://www.ck12.org
Development Team
Writer: Jicelle E. Gegrimos Malubal National High School
Editor/QA: Eugenio E. Balasabas Ressme M. Bulay-og Mary Jane I. Yeban
Reviewer: Gina I. Lihao EPS-Mathematics
Illustrator: Layout Artist:
Management Team: Evelyn F. Importante OIC-CID Chief EPS Jerry c. Bokingkito OIC-Assistant SDS Aurelio A. Santisas, CESE OIC- Assistant SDS Jenelyn A. Aleman, CESO VI OIC- Schools Division Superintendent