Verifying the Triangle� Proportionality Theorem

FOR MATH MAJORS

1. 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)  

2. RECALL: “PERFECTLY PROPORTIONAL: Are We Meant To Be?” Determine whether the two ratios form a proportion. Use cross multiplication to verify your answer. 2. 15 : 12 = 4 : 5 Solution: : = 15 (5) = 12 (4) 75 ≠ 48 (NO) 3. 4 : 6 = 2 : 3  

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

4. Activity: Verifying the Triangle Proportionally Theorem 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

5. Activity: Verifying the Triangle Proportionally Theorem Procedure: 4. Calculate Ratios: Compute the ratio of: 5. Analyze and Conclude: Compare the two ratios (if they are equal, the Triangle Proportionally Theorem is verified)  

6. A D E B C

7. AD = 4 DB = 6 AE = 2 EC = 3

8. RATIO: = = ÷ = b ) = … =  

9. TRIANGLE PROPORTIONALITY THEOREM If a line is parallel to one side of a triangle intersects the other two sides of the triangle, then the line divides these two sides proportionally.

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

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

12. THE TRIANGLE TRACKER: Find the Missing Segment! Calculate the missing segments in the figures below. 2. 12 y 4 8 7 x

13. THE TRIANGLE TRACKER: Find the Missing Segment! Calculate the missing segments in the figures below. 1. 3 24 x y 15 5

14. 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.

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

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

17. III. Solve x: x 5 8 2

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

19. Quiz: I. Tell whether the proportion is Right or Wrong. I. L M K N O = = =  

20. II. Find the uknown quantity. 10 x 14 18

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

22. 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.

23. 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.

24. 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.

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