1. I will Measure You: It Matters not, how Tall are You Prepared by: Rolando Jaca Amy Caballero Lea Buligao

2. Brief Description of Thales The Greek philosopher, Thales was born around 624BC, the son of Examyes and Cleobuline. He came from a distinguished family. He was an engineer, scientist, mathematician, and philosopher, the first natural philosopher in the Milesian School..

3. His contribution to math Thales was known for his innovative use of geometry. His understanding was theoretical as well as practical. For example, he said: “Space is the greatest thing, as it contains all things”. Thales understood similar triangles and right triangles, and what is more, used that knowledge in practical ways.

4. How Thales found the height of the pyramid? The story is told that he measured the height of the pyramids by their shadows at the moment when his own shadow was equal to his height. A right triangle with two equal legs is a 45-degree right triangle, all of which are similar. The length of the pyramid’s shadow measured from the center of the pyramid at that moment must have been equal to its height.

5. The modeled situation It is about the height of the pyramid with the help of the pyramid shadow and a stick.

6. The pyramid The pyramid base, a parallelogram, is designed using the center property with central symmetry. The pyramid height is designed to have a perpendicular direction to the front side of the base. The stick is parallel to the pyramid height. Dashed segments are used to represent the hidden sides of the pyramid. An arc is used to represent the sun position.

7. The shadow The sun is modelized as a point on the arc. The pyramid shadow is constructed semi-arbitrary with the ray going from the sun position to the pyramid summit. An arbitrary point in this ray is chosen as the shadow of the pyramid summit. The shadow of the pyramid height is constructed from this point and the pyramid base center.

8. Definition of similar triangles Triangles are similar if they have the same shape, but can be different sizes.

9. Two triangles are similar iff: all the corresponding angles are equal, or all the corresponding sides are proportional. angle A = angle D angle B = angle E angle C = angle F AB/DE = BC/EF = AC/DF

10. Shadow Method Using shadows is a quick way to estimate the heights of trees, flagpoles, buildings, and other tall objects. To begin, pick an object whose height may be impractical to measure, and then measure the length of the shadow your object casts. Also measure the shadow cast at the same time of day by a yardstick (or some other object of known height) standing straight up on the ground.

11. Shadow Method TIME CAPTURED: Wednesday, February24, 2010 12:30 pm in front of CED Bldg. (MSU-IIT) TIME CAPTURED: Wednesday, February24, 2010 12:30 pm in front of CED Bldg. (MSU-IIT)

12. DIAGRAM h h 109 in. 64 in 34 in. Tree’s shadow Jaca’s shadow TIME CAPTURED: Wednesday, February24, 2010 12:30 pm in front of CED Bldg. (MSU-IIT) TIME CAPTURED: Wednesday, February24, 2010 12:30 pm in front of CED Bldg. (MSU-IIT)

13. To show that the two triangles are similar

14. Solving for h Using triangle similarity by ratio and proportion and let h be the height of the tree then we have: 64 in h 34 in 109 in 34 in(h) = 64 in (109in) 34 in(h) = 6976 in 2 34 in(h) 6976 in 2 34 in 34 in Therefore the height of the tree is 205.18 in h = 205.18 in.

15. Mirror Method Choose a tall object with a height that would be difficult to measure directly, such as a football goalpost, a basketball hoop, a flagpole.

16. Steps: Step 1 Mark crosshairs on your mirror. Use tape or a soluble pen. Call the intersection point X. Place the mirror on the ground several meters from your object. Step 2 An observer should move to a point P in line with the object and the mirror in order to see the reflection of an identifiable point F at the top of the object at point X on the mirror. Make a sketch of your setup, like this one. Step 3 Measure the distance PX and the distance from X to a point B at the base of the object directly below F. Measure the distance from P to the observer’s eye level, E.

17. Step 4 Think of FXas a light ray that bounces back to the observer’s eye along XE. Why is B P? Name two similar triangles. Tell why they are similar. Step 5 Set up a proportion using corresponding sides of similar triangles. Use it to calculate FB, the approximate height of the tall object. Step 6 Write a summary of what you and your group did in this investigation. Discuss possible causes for error.

18. Mirror Method Time captured: Wednesday, February24, 2010 03:10 pm in front of gym bldg. (MSU-IIT)

19. mirror post Leah y 8.9 in 41 in. 62.5 in DIAGRAM

20. To show that the two triangles are similar Figure 1 A B C D E

21. To show that the two triangles are similar Figure 2 A C D E B

22. Solving for y Using triangle similarity by ratio and proportion and let y be the height of the post then we have: y 62.5 in. 89 in. 41 in 41in.(y) = 3.062 in 2 41in.(y) 3.062 in 2 41in. 41in. Therefore the height of the post is 74.61 in. y= 74.61 in.

23. Clinometers The method of measurement requires a protractor (clinometer), a straw and a measuring tape. This method does not use the shadows but uses accurate visual senses of the measurer. The protractor preferably must be as big as possible for accurate angle observation and for further calculation.

24. Material Required 1-Protactor 2-Straw 3-Stone 4-Thread

25. Procedure Step 1 Get a protractor with one straight edge (a 180 degree protractor). Step 2 Tape a straw along the straight edge of the protractor.

26. Step 3. Tie a string through the small hole on the straight edge that is directly across from the 0 degree mark on the protractor. This may also be labelled as 90 degrees. If your protractor does not have a small hole here, or if the hole is not situated correctly (this is a common problem with some cheap protractors), tape or glue the string to the protractor at this mark. Make sure the string dangles a few inches below the protractor.

27. Step 4 Attach a washer or fishing weight to the dangling end of the string. Step 5 Sight the top of a tall object through the straw.

28. Step 6 Note the number where the string crosses. Subtract this number from 90 to determine the angle of elevation between your eye and the top of the object you are sighting Reminders It is necessary to add our height to the calculated height of the object as our reference point (eye level) is above the ground. It is not necessary to add our height if, we measure the angle keeping our eye at ground level (on which the building stands).

29. Clinometers Time captured: Wednesday, February24, 2010 4:12 pm in front of Asnie boarding house (Tibanga I.C.) Time captured: Wednesday, February24, 2010 4:12 pm in front of Asnie boarding house (Tibanga I.C.)

30. Aha!!!!!! The string crosses at 35 0,so I should subtract it from 90 0 according to the rule to get the angle of elevation between my eye and the top of the electric post. Then, I have: 90 0 -35 0 = 55 0. Therefore the angle of elevation is 55 0. Aha!!!!!! The string crosses at 35 0,so I should subtract it from 90 0 according to the rule to get the angle of elevation between my eye and the top of the electric post. Then, I have: 90 0 -35 0 = 55 0. Therefore the angle of elevation is 55 0.

31. 55 0 z z 157 in. Electric post roland Time captured: Wednesday, February24, 2010 4:12 pm in front of Asnie boarding house (Tibanga I.C.) Time captured: Wednesday, February24, 2010 4:12 pm in front of Asnie boarding house (Tibanga I.C.) DIAGRAM 157 in. Angle of elevation 64 in.

32. Solving for z Using tangent function we have: tan 55 0 = z /157 in. z = tan 55 0 (157 in. z = 1.43 (157 in.) z = 224.22 in.

33. To find the height of the electric post: As the reminders remind me. I should add my height which is 64 in. from the value of z which is 224.22 because my eye is sighting not on the level ground. So the height of the electric post is: Z + 64 in. = 224.22 in + 64 in. = 288.22 in.

34. Still on the ground but still can’t out of reach Time captured: Wednesday, February24, 2010 4:12 pm in front of Gym Building (MSU-IIT)

35. Cherie Leah Amy Jessie Asnie 111 in. x x 25 in. 202 in. DIAGRAM

36. Solving for x Using triangle similarity by ratio and proportion and let x be the length across the rotunda then we have: 202 in. X+111 in. 25 in. 111 in. (25 in.)(x+111 in.) = (111 in.)(202 in.) 25 in. (x )+ 2775 in 2. = 22422 in 2. 25 in. (x ) = 22422 in 2. - 2775 in 2. 25 in. (x ) = 19647 in 2. 25 in. (x ) 19647 in 2. 25 in. 25 in. Therefore the length across the rotunda is 785.88 in. x = 785.88 in. x = 785.88 in.

37. The hunters

38. BABAY U!!!!!!!!!!!!!!!! INGATZ thanks for listening