Proportions & Ratios

1. Proportions & Ratios Objective: In this section students will learn to use proportions and similar figures to: solve problems involving geometric figures and to measure objects indirectly.

2. Similar Figures: have exactly the same shape but not necessarily the same size. A Definition of Similar Figures

3. Corresponding sides of two figures are in the same relative position A Definition of Corresponding Sides

4. Corresponding Angles are in the same relative position. Two figures are similar if and only if the lengths of corresponding sides are proportional and all pairs of corresponding angles have equal measures. A Definition of Corresponding Angles

5. When stating that two figures are similar, use the symbol Application of Proportions

6. A recipe calls for 2 cups of sugar for each 5 cups of flour. How many cups of sugar are needed for 15 cups of flour? Cake Recipe and Proportions

7. 6 cups of sugar are needed for 15 cups of flour. Writing a proportion is a useful method in solving word problems. Be careful to write the terms of the proportion in the order stated in the problem.

8. A light green paint is made by mixing 8 parts of white paint to 3 parts of green. How many quarts of green paint are mixed with 24 quarts of white to make the light green color? 9 quarts of green paint are mixed with 24 quarts of white.

9. A totem pole casts a shadow 45 feet long at the same time that a 6-foot-tall man casts a shadow that is 3 feet long. Write and solve a proportion to find the height of the totem pole.

10. Both the man and the totem pole form right angles with the ground, and their shadows are cast at the same angle. Therefore you can form two similar right triangles The totem pole is 90 ft. tall.

11. Indirect Measurement Ex. 1 Indirect measurement is a method of using proportions to find an unknown length or distance in similar figures. Two common ways to achieve indirect measurement involve: using shadow lengths to find an object's height. using a mirror on the ground to obtain the height of an object.

12. How tall is the flagpole? Indirect Measurement Ex. 2

13. Indirect Measurement Practice 1 A 6.5 ft. tall car standing next to an adult elephant casts a 33.2 ft. shadow. If the adult elephant casts a shadow that is 51.5 ft. long then how tall is it?

14. Indirect Measurement Practice 2 A map has a scale of 3 cm : 18 km. If Riverside and Smithville are 54 km apart then they are how far apart on the map?

15. A 6 ft. tall tent standing next to a cardboard box casts a 9 ft. shadow. If the cardboard box casts a shadow that is 6 ft. long then how tall is it? Indirect Measurement Practice 3

16. If a 42.9 ft. tall new lunar flagpole casts a 253.1 ft. long shadow then how long is the shadow that a 6.2 ft. tall astronaut casts? Indirect Measurement Practice 4

17. The Eiffel Tower does not have a scale written on its side. However, we can still measure this structure using our knowledge of similar figures. Similar triangles can be used also to indirectly measure this structure.

18. Measuring the Eiffel Tower in Three Steps 1. measures out a point 500 meters from the base of the tower, and places a small mirror flat on the ground. 2. Stand behind the mirror in such a spot that standing upright one sees the top of the tower reflected in the mirror. 3. Measures both the distance from the spot where one stands to the mirror (2.75 meters) and the height of one’s eyes from the ground (1.8 meters).

19. The Law of Reflection states, “The angle at which the light reflects off the mirror is the same as the angle at which it hits the mirror.” Using this principle and the figure above, you can conclude that these triangles are similar with proportional sides. The Law of Reflection

20. 500m 2.75m The Eiffel Tower, according to this calculation, is approximately 327.3 meters high. The Results of Our Indirect Measurement This means that the ratio of the long leg in the large triangle to the length of the long leg in the small triangle is the same ratio as the length of the short leg in the large triangle to the length of the short leg in the small triangle.

21. Proportions and Blueprints Someday you may need to interpret blueprints for a new project. You will need a basic understanding of symbols and proportions to help avoid confusion throughout the process and attain your vision of the project.

22. An architect is a person who plans, designs, and oversees the construction of buildings.

23. Cities are beginning to change. More and more creative young people are drawn to progressive cities every year. Many of the brightest of the next generation want to participate in making a better future. Additionally, over the next 40 years, our population will expand unlike anything humanity has ever seen and our world’s cities swell by billions. All of this growth equals opportunity. How we build our cities determines how we use energy within them. Denser, more walkable communities use much less energy than car-dependent ones. Multifamily homes use much less than homes on big lots. Compact urban infrastructure beats sprawling systems. Even consumer choices change in compact communities. For example, how many condo owners have home gyms? Climate-focused city planning can lead to massive reductions in per capita energy use. Which will, in turn, spur rapid economic growth. With good climate-focused city planning and a commitment to urban innovation, cities will begin to revitalize neighborhoods, prepare local businesses for global competition and rising energy costs, and become magnets for talent and new thinking. A hundred cities committed to carbon-zero futures would be a hundred cities on their way back to prosperity — and a brighter future for the planet. “Decide what to be and go be it”

24. MANHATTAN - 2050

26. BARCELONA, SPAIN - 2050

27. BEIJING, CHINA - 2050

28. Beirut, Lebanon - 2050

29. Florence, Italy - 2050

30. London, UK - 2050

31. Paris, France - 2050

32. Mumbai, India - 2050

33. Rio de Janeiro, Brazil - 2050

34. Brooklyn – Right F#%&@’n Now

35. All of the information is drawn to scale. • Every one-quarter or one-eighth of an inch on the blueprint equals one foot in actual size. • On plans drawn to a one-quarter of an inch scale, for instance, two inches equal eight feet in real life. • Each page of the drawings will be labeled to indicate what size scale the architect or home designer used when drafting the plans. • You can, with the help of similar figures and proportions, calculate the conversions yourself. How to Interpret a Blueprint

37. Helpful Conversions to Know