CH 1: BasicS of Geometry

1. CH 1: BasicSof Geometry Ms. Vasili’s Review with real life facts

2. Where do you see mirrors in your world? • Our exploration of mirrors begins with a simple principle of light reflection. Mirrors, and other reflective surfaces, reflect light in a very predictable manner. The angle at which light approaches the surface is the same as the angle at which light bounces off the surface. This timeless truth regarding light is referred to as the law of reflection.

3. Where do you see mirrors in your world? • We often think of light as a ray traveling in a straight line through a uniform medium. Physicists often use laser beams to model the behavior of these light rays. Here we observe light bouncing off a collection of mirrors before finally exiting the arrangement. Each reflection occurs according to the law of reflection

5. Where do you see mirrors in your world? • Think about it. There's mirrors on cars to see your blind spot. There's mirrors on bathroom walls to see your bald spot. There's mirrors in department stores to prevent five-finger discounts. And there's mirrors in the periscope of your family submarine that serves as a lookout. You'll find mirrors in many other places, but for now let's talk about those periscopes.

6. Continue • Periscopes are equipped with two flat or plane mirrors that are angled at a 45° angle to the column of the periscope. Light enters through the opening in the top of the periscope and reflects off the first mirror. The reflected light is channeled down the column where it reflects off the second mirror and travels to the observer's eye. By aligning these two mirrors at just the right angle you can spot your neighbor's battleship before shooting it with a torpedo. Now that's physics for better living.

8. Curved mirrors have very practical applications in the building of satellite dishes and radio telescopes. The photo that we see represents the Parkes Radio Telescope in Australia. The reflecting surface measures 64 meters across (nearly 200 feet). Its large area allows it to capture more radio light waves. All concave reflectors are able to collect light from a distant source and focus it at a point known as a focal point. In the case of a radio antenna, the receiver is placed at the focal point in order to collect all this reflected light. The overall process is capable of making a weak signal strong enough to decipher.

9. Continue Satellite dishes and radio telescopes often use curved mirror surfaces that have a parabolic shape as opposed to a spherical shape. Large spherical reflectors have a drawback of not focusing all the collected light to a single point. This is often referred to as spherical aberration. Parabolic surfaces correct for this defect and are quite capable of producing a clear signal of the light waves that reflect off its entire surface.

11. Patterns and Inductive Reasoning

12. What do You Know about Inductive Reasoning? • Look for a pattern • Make a conjecture • Verify a conjecture Looking for patterns and making conjectures is part of a process called Inductive Reasoning

13. Points , Lines, and Planes Astronomers usually name different constellations in the night by looking at the figure formed when segments are drawn to connect some or all stars that make up the constellation.

14. Continue Virgo is the second-largest constellation in the sky. It is depicted as an angel with wings, holding an ear of wheat. She is usually identified as Dike, the goddess of justice, who was the daughter of Zeus and Themis. She lived on Earth during the Golden Age of mankind. This was a time of peace and happiness where it was always springtime and food grew without cultivation and humans never grew old. The men lived like gods, not knowing work, sorrow or war.

15. Continue • Zeus overthrew his father, Cronus, on Olympus and the Silver Age began. Zeus shortened spring time and introduced the yearly cycle of seasons. People during this time became quarrelsome and no longer honored the gods. Virgo longed for the Golden Age times and assembled the human race together and spoke sternly to them about changing their ways. She warned them that "worse was to come". Afterwards, she spread her wings and took refuge in the mountains, turning her back on mankind. • The Age of Bronze and Iron fell upon mankind. Humans descended into violence, theft and war. Virgo could no longer bear the sins of humanity and abandoned the Earth and flew up to heaven. To this day, she sits next to the constellation of Libra, which is seen as the scales of justice.

16. Continue Gerrit Rietveld was born in Utrecht, Netherlands, where he lived and worked all his life.  After learning cabinet making from his father, he trained as an architectural draftsman, becoming an architect in 1919.  Rietveld joined De Stijl, a Dutch Modernist design movement which held fast to rules relating colors to their symbolic meanings, promoting primary colors and simple forms and tried to reduce objects to their essential forms. The artist used red and blue rectangle to represent planes extending into space. The black bar represented lines through space. Rietveld used yellow squares to represent points in space.

17. What do you Know about Points, Lines, and Planes?

18. Space – a boundless, 3-dimensional set of all points. Collinear– points that lie on the same line Coplanar - points that lie on the same plane

19. 2 lines intersect at a point. 2 planes intersect in a line.

20. Segments and their measures Fabric is measure in bolts. A bolt, which is 120 feet long , is only one of many units of linear measure used in the world. A piece of the thread running the length of the fabric could be a model for a line segment.

21. Midpoint Formula • Things to remember  • A(xa,ya) and B(xb,yb) are two points in the plane, and P(xp,yp) is the midpoint of the line joining A and B: • The coordinates of the midpoint ‘P’ are given by: and

22. Distance Formula • Given the two points (x1, y1) and (x2, y2), the distance between these points is given by the formula: • Remember relation: Pythagorean Theorem- Distance Formula

23. Money Lines and Angles In 1914 , THE SECRET SERVICE SEIZED $45.7 million in counterfeit bills before they entered U.S. circulation-though forgers sneaked $25.3 million in fakes into the economy…. The potential of desktop counterfeiting has guided the design, materials, and the production decision of the next generation of U.S. currency…. The new currency sports several security-enhancing features…. An enlarged, off -center portrait…increases recognition, reduces wear, and open space for a watermark; extra detail stymies duplication…Numerals printed in color –shifting ink appear green when viewed straight an and black when see from an angle…

24. Angles and their measures The Japanese have long appreciated the beauty of nature through the art of ikebana. In ikebana, flowers and branches are arranged in a simple way that allows the beauty of each piece to be seen clearly. When creating an arrangement, the placement of each branch or flower is determined by forming an angle of a specific size.

25. Angle pair relationship SKI JUMPING In order for ski jumpers to achieve the maximum distance on a jump, they need to make the angle between their body and the front of their skis as small as possible. This allows them to get the proper extension over the tips of their skis.

26. Continue • Aviation • Compass headings are used to indicate the directions of airplanes. A heading is stated as the measure of the angle formed by the flight path of the airplane and an imaginary path in the direction of due north. • A pilot is on approach to land at a compass heading of 68⁰ NW. The tower has instructed her to land on runway 9, which has a heading of 90⁰ NW. How many degrees must the pilot turn her plane in order to land?

27. What do You Know about Angles?

28. Architecture, Landscaping

29. Perimeter, Area, Circumference

30. Resources: http://www.flickr.com