Verifying the Triangle� Proportionality Theorem

for math majors

1. OPENING PRAYER Oh God, Almighty, behold us Thy loving children, offering Thee today, our works and studies. Help us dear Lord, to be obedient to our teachers, kind to our companions, diligent in our works and studies, so that we may merit Thy blessings to ourselves, to our school and to our beloved country, the Philippines. Amen.

2. Ipad : Jose went to the grocery store. He bought 20 packs of cookies, 15 packs of noodles and 12 packs of 3-in-1 coffee. How many packs of groceries did he buy in all? Let x = be the total number of packs of groceries bought by Jose. x = 20 packs of cookies + packs of noodles + 12 packs 3-in-1 coffee. x = 47 packs of groceries

3. RECALL: RECALL: “PERFECTLY PROPORTIONAL: Are We Meant To Be?” Determine whether the two ratios form a proportion. Use cross multiplication to verify your answer. 3 : 5 = 6 : 10 Solution: = 3(10) = 5 (6) 30 = 30 (YES)  

4. RECALL: RECALL: 2. 15 : 12 = 4 : 5 Solution: : = 15 (5) = 12 (4) 75 ≠ 48 (NO)  

5. RECALL: RECALL: 3. 8 : 12 = 2 : 3

6. RECALL: Activity: Verifying the Triangle Proportionality Theorem Objective: To demonstrate that a line parallel to one side of a triangle divides the other two sides proportionally. Materials: ruler, protractor, graphing paper, and pencil

7. RECALL: Procedure: Construct a triangle: Using a ruler, draw triangle ABC on graphing paper. Label the vertices as A, B, and C. Draw a parallel line: Select a point D on side AB and a point E on side AC such that DE is parallel to BC (ensure accuracy using a protractor). Measure and Record Lengths: Measure segments AD, DB, AE, and EC using a ruler.

8. RECALL: 4. Calculate Ratios: Compute the ratios of: 5. Analyze and Conclude: Compare the two ratios (If they are equal, the Triangle Proportionally Theorem is verified).  

9. Let AD = 4 DB = 6 AE = 2 EC = 3

10. RATIO: = = ÷ = = Therefore, DE BC b ) =  

11. PROVE IT OR LOSE IT: Are They Parallel? Determine whether a segment is parallel to one side of a triangle. 1. C 12 8 K L 24 16 A E Is KL AE?

12. PROVE IT OR LOSE IT: Are They Parallel? Determine whether a segment is parallel to one of a triangle. 2. P 28 30 C D 10 11 R T Is CD RT ?

13. THE TRIANGLE TRACKER: Find the Missing Segment! Calculate the missing segments in the figures below. 1.. 12 y 4 8 7 x

14. THE TRIANGLE TRACKER: Find the Missing Segment! Calculate the missing segments in the figures below. 2. 3 24 x y 15 5

15. Team Sync: LET’S VOLT IN! A X-2 I F X-3 T 12 9 H Find x so that TI // FA. Find the measures of FT and AI.

16. I. Identify whether the proportion is correct-write True or False. I H R G T = = =  

17. II. Supply the missing term to complete the proportion. L A E N R = = = ?  

18. III. Solve for x: x 5 8 2

19. Key Points About Real-life Applications of the Triangle Proportionality Theorem

20. Architecture and Construction Architects and builders use this theorem to ensure proper proportions when designing buildings and structures, calculating the lengths of support beams or determining precise measurements based on parallel lines and similar triangles.

21. Surveying Land surveyors can utilize the triangle proportionality theorem to calculate distances between points on the ground by creating similar triangles using unknown reference points and prallel lines.

22. Indirect Measurement By utilizing the principle of parallel lines and similar triangles, one can calculate the height of a tall object (like a tree) by measuring its shadow and comparing it to the shadow of a known object.

23. Map Reading When interpreting maps, the concept of similar triangles based on the proportionality theorem can be used to estimate real-world distances based on the scale of the map.

24. What is Triangle Proportionality Theorem? If a line parallel to one side of a triangle intersects the other two sides, then it divides those sides proportionally.

25. Quiz: I. Evaluate whether the proportion is right or wrong. L I. M K N O = = =  

26. II. Find the uknown quantity. 10 X 14 18

27. CLOSING PRAYER Dear God, the Giver of all things, thank you for all the blessings that You had given us. I'm sorry for all my faults and humbly ask for your forgiveness. Bless all my teachers and schoolmates. Teach us all to love one another and to love you above all things. Amen.