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GEOMETRY

GEOMETRY. GEOMETRY. GEOMETRY is about measuring. GEO means “Earth” METRY means “Measure”. GEOMETRY is also about shapes, their properties and relationships. G E O M E T R I C I L L U S I O N S. 1.1. Optical Illusions through art using geometric concepts.

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GEOMETRY

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  1. GEOMETRY

  2. GEOMETRY GEOMETRY is about measuring GEO means “Earth” METRY means “Measure” GEOMETRY is also about shapes, their properties and relationships

  3. GEOMETRIC ILLUSIONS 1.1 Optical Illusions through art using geometric concepts

  4. GEOMETRIC ILLUSIONS 1.1 Penrose triangle The Penrose triangle, also known as the Penrose tribar, is an impossible object. The impossible cube or irrational cube is an impossible object that draws upon the ambiguity present in a Necker cube illusion. An impossible cube is usually rendered as a Necker cube in which the edges are apparently solid beams. This apparent solidity gives the impossible cube greater visual ambiguity than the Necker cube, which is less likely to be perceived as an impossible object. The illusion plays on the human eye’s interpretation of 2-dimensional as 3-dimensionall objects.

  5. GEOMETRIC ILLUSIONS 1.1 A famous perceptual illusion in which the brain switches between seeing a young girl and an old woman

  6. GEOMETRIC ILLUSIONS 1.1 The Kanizsa triangleis an optical illusions first described by the Italian psychologist GaetanoKanizsa in 1955. In the accompanying figure a white equilateral triangle is perceived, but in fact none is drawn. This effect is known as a subjective or illusory contour. Also, the nonexistent white triangle appears to be brighter than the surrounding area, but in fact it has the same brightness as the background.

  7. GEOMETRIC ILLUSIONS The Necker cube is an ambiguous line drawing. It is a wire-frame drawing of a cube in isometric perspective, which means that parallel edges of the cube are drawn as parallel lines in the picture. When two lines cross, the picture does not show which is in front and which is behind. This makes the picture ambiguous; it can be interpreted two different ways. When a person stares at the picture, it will often seem to flip back and forth between the two valid interpretations (so-called multistable perception).

  8. 1.3 Identifying Congruent and Similar Figures Congruent means that the figures must have same shape and size Similar means figures must have same shape, but can also have same size

  9. 1.4 Exploring Symmetry A figure has Line Symmetry or Reflection Symmetry if it can be divided into 2 parts, each of which is the mirror image of the other. Some figures have 2 or more lines of symmetry. a. b. c. This lobster figure has a horizontal line of symmetry. This saucer figure has a vertical and horizontal line of symmetry This ying/yang figure has 2 congruent parts, but it has no line of symmetry Although this ying/yang figure lacks line symmetry, it does have Rotational Symmetry. To be rotational a figure must coincide with itself after rotating 180⁰ or less, either clockwise or counter-clockwise.

  10. 1.4 Exploring Symmetry Reflective Symmetry Rotational Symmetry Line of Symmetry Mirror image of the other side of line of symmetry Must coincide with itself after rotating 180 ⁰or less

  11. 1.4 The Midpoint Formula • A ( ─2, 5) • The midpoint of the line segment joining A (x1, y1) and B (x2, y2) is as follows: C ( 1, 3) • B ( 4, 1) Each coordinate of M is the mean of the corresponding coordinates of A and B.

  12. To understand GEOMETRY you must first learn to speak the language by studying the terminology Next you need to understand the logic of geometry through deductive and inductive reasoning Then you need to practice applying learned principles in new problem situations Finally you need to remember learned concepts as building blocks for new ones to follow

  13. The Building Blocks of Geometry UNDEFINED TERMS : can’t be defined by simpler terms. [ Point , Line, Plane ] DEFINED TERMS : can be defined by the undefined terms or previously defined terms so it is easier to describe geometric figures and relationships POSTULATES : a statement that is accepted without proof. THEOREMS : a statement that is proven to be true. The progression of undefined, defined terms, postulates and theorems lead to evolving new logical generalizations.

  14. 1.5 Coordinate & Noncoordinate Geometry Coordinate Plane y axis Quadrant II ( – , + ) Quadrant I ( + , + ) ( 0 , 0 ) x axis Quadrant III ( – , – ) Quadrant IV ( + ,– ) x coordinate is first y coordinate is second Ordered pairs in form of ( x , y )

  15. 1.5 Slope and Rate of Change SLOPEof a non-vertical line is the ratio of a vertical change (RISE) to a horizontal change (RUN). Slope of a line: m = y2 – y1 = RISE x2 – x1= RUN (differences in y values) (differences in x values) y ( x2,y2) •  y2 – y1 RISE ( x1,y1) • x2 – x1 RUN x

  16. 1.5 Slope and Rate of Change CLASSIFICATION OF LINES BY SLOPE A line with a +slope rises from left to right [ m > 0 ] Positive Slope A line with a –slope falls from left to right [ m < 0 ] Negative Slope A line with a slope of 0 is horizontal [ m = 0 ] 0 Slope A line with an undefined slope is vertical [ m = undefined, no slope ] No Slope

  17. 1.5 Slope and Rate of Change SLOPES OF PARALLEL & PERPENDICULAR LINES PARALLEL LINES: the lines are parallel if and only if they have the SAME SLOPE. m1 = m2 PERPENDICULAR LINES: the lines are perpendicular if and only if their SLOPES are NEGATIVE RECIPROCALS. m1 = - 1 m2 or m1 m2 = - 1

  18. Slope of a line: m = y2 – y1 = RISE (differences in y values) x2 – x1= RUN (differences in x values) 1.5 Slope and Rate of Change Ex 1: Find the slope of a line passing through ( – 3, 5 ) and ( 2, 1 ) Let ( x1, y1 ) = ( – 3, 5 ) and ( x2, y2 ) = ( 2, 1 ) m = 5 – 1 = 4 – 3 – 2 – 5 OR m = 1 – 5 = – 4 2 + 3 5 5 • ( - 3, 5 ) –4 • x ( 2, 1 ) y

  19. Coordinate & Noncoordinate Geometry Coordinate Geometry (also known as Analytic Geometry) Shows a graph Non-Coordinate Geometry (also known as Euclidean Geometry) has no graph

  20. 1.6 Perimeter & Area Square Rectangle P = 4 s and A = s2 P = 2 l + 2 w and A = l w Triangle Circle P = a + b + c and A = ½ b h C = 2 ∏ r and A = ∏ r2 w l S S r a c ● h b

  21. 1.6 Find the Perimeter & Area Example 1 Each square on the grid at the left is 1 foot by 1 foot. Find the perimeter and area of the green region. SOLUTION: The distance around the shaded figure is 40 units. Thus the perimeter of the region is 40 feet. By counting the shaded squares, you can find that the area is 36 square feet. Can you think of another way to find the area? Subtract the number of non-green units within the 6 x 8 = 48 total units. Or, 48 – 12 = 36

  22. 1.6 Finding an Area 6 ft What is the area of the sidewalk? 6 ft 40 ft Area of sidewalk Area of larger rectangle Area of pool = ─ = ( 36 ) ( 52 ) – ( 24 ) ( 40 ) = 1872 – 960 = 912 square feet 24 ft

  23. 2.1 Undefined Terms • • P

  24. 2.1 This illustration shows that lines may lie in different planes or in the same plane. B L P ● B ● k A Here lines L and AB are on plane “B” while lines k and AB are on plane “P”. NOTE: line AB lies in both “B and P” planes, which is also a intersecting line of these planes.

  25. 2.1 Defined Terms • • B A ● ● L M ● ● ● X K B

  26. 2.1 Defined Terms J ● ● ● K L Vertex: K Sides KJ and KL 3 < 1 and < 2 are adjacent < 1 and < 3 are not adjacent 2 ● 1 V

  27. 2.1 Naming Angles R R ● ● Exterior of angle Interior of angle ) 1 ● ● T T ● ● Exterior of angle B B

  28. 2.1 Naming Angles B C In order to uniquely identify the angle having D as its vertex, we must either name the angle using three letters or introduce a number into the diagram. ) ) ) A D Figure 3

  29. 2.1 Example of Naming Angles Use three letters to name each of the numbered angles in the accompanying diagram. M B C 3 E 2 < 1 = < 2 = < 3 = < 4 = 1 4 A D L

  30. 2.1 Example of Naming Lines and Line Segments ● ● ● J W R

  31. 2.1 Definitions • The purpose of a definition is to make the meaning of a term clear. A good definition must: • Clearly identify the word or expression that is being defined • State the distinguishing characteristics of the term being defined, using commonly understood or previously used • Be expressed in a grammatically correct sentence • Example: consider the terms collinear and non-collinear points • Collinear points are points that lie on the same line • Non-collinear points are points that do not lie on the same line ● S ● ● ● C R ● B ● A T Much of geometry involves building on previously discussed ideas. Example: we can use current knowledge of geometric terms to arrive at a definition of a triangle. How would you draw a triangle? If you start with 3 non-collinear points and connect them with line segments, a triangle is formed.

  32. 2.1 Definitions A good definition must be reversible as shown in the following table The first two definitions are reversible since the reverse of the definition in a true statement. The reverse of the third definition is false since the points may be scattered as in example to right. ● ● ● ● ● ●

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