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