Chapter 9: Geometry

9.1: Points, Lines, Planes and Angles 9.2: Curves, Polygons and Circles 9.3: Triangles (Pythagoras’ Theorem) 9.4: Perimeter, Area and Circumference 9.6: Transformational Geometry 9.7: Non-Euclidean Geometry 9.8 Chaos and Fractals

Points, Lines and Angles Line AB Half-line AB Ray AB Segment AB Angle ABC 9.1

Types of Angles Acute Right Obtuse Straight Complementary Supplementary 9.1

Vertical Angles Vertical angles have equal measure 9.1

More Angles Which angles are equal? 9.1

Curves Simple Closed 9.2

Convex and Concave 9.2

Polygons A polygon is a simple, closed curve made up of straight lines. A regular polygon is convex with all sides equal and all angles equal. 9.2

Triangles Angles: acute, right or obtuse Sides: equilateral, isosceles, scalene 9.2

Angle Sum of Triangle 9.2

Quadrilaterals Trapezoid Parallelogram Rectangle Square Rhombus 9.2

Circle Center Radius Chord Diameter Semicircle Tangent Secant 9.2

Pythagoras’ Theorem For a right triangle, a 2 + b 2 = c 2 9.3

Proof of Pythagoras Add up the area of the big square two ways: one big square or 4 triangles plus one smaller square 9.3

#76 Depth of Pond There grows in the middle of a circular pond 10 feet in diameter a reed that projects one foot out of the water. When it is drawn down, it just reaches the end of the pond. How deep is the water?

Perimeter The perimeter of a plane figure composed of line segments is the sum of the measures of the line segments, so the total length around the object. It is measured in linear units. 9.4

Area The area of a plane figure is the measure of the surface covered by the figure. 9.4

Perimeter of a Triangle Triangle with sides of length a, b, and c has P = a + b + c 9.4

Area of Triangle Triangle with base b and height h A = ½ bh 9.4

Perimeter and Area of Rectangle Rectangle with length l and width w has P = 2l + 2w = 2(l + w) A = lw 9.3

Perimeter and Area of Square If all sides have length s, then P = 4s A = s 2 9.3

Area of Parallelogram Parallelogram with height h and base b A = bh 9.3

Area of Trapezoid Trapezoid with parallel bases b and B and height h A = ½ h (b + B) 9.4

Circumference and Area of Circle Circle of radius r has circumference C = πd = 2πr And area A = πr 2 9.4

Euclid’s Postulates 1.Two points determine one and only one straight line segment 2.A straight line extends indefinitely far in either direction 3. A circle may be drawn with any given center and any given radius 4. All right angles are equal 5. Given a line k and a point P not on the line, there exists one and only one line m through P that is parallel to k 9.7

Non-Euclidean Geometry The first four postulates are much simpler than the fifth, and for many years it was thought that the fifth could be derived from the first four It was finally proven that the fifth postulate is an axiom and is consistent with the first four, but NOT necessary (took more than 2000 years!) Saccheri ( ) made the most dedicated attempt with his quadrilateral Any geometry in which the fifth postulate is changed is a non-Euclidean geometry 9.7

Saccheri’s Quadrilateral He assumed angles A and B to be right angles and sides AD and BC to be equal. His plan was to show that the angles C and D couldn’t both be obtuse or both be acute and hence are right angles.

Riemannian (Spherical) Geometry 5 th : Through a point P off a line k, no line can be drawn that is parallel to k. Lines have finite length. Angles in Saccheri’s quadrilateral are obtuse. Angles in triangle sum to more than

Lobachevskian (Hyperbolic) Geometry 5 th : Through a point P off the line k, at least two different lines can be drawn parallel to k Lines have infinite length Angles in Saccheri’s quadrilateral are acute Angles in triangle sum to less than

Fractals What do we mean by dimension? Consider what happens when you divide a line segment in two on a figure. How many smaller versions do you get? Consider a line segment, a square and a cube. 9.8

Self-similarity An object is self-similar if it can be formed from smaller versions of itself (with no gaps or overlap) A square is self-similar, a circle is not. Many objects in nature have self-similarity. 9.8

More self-similarity in Nature 9.8

Self-similar fractals 9.8

Sierpinski Gasket Start with an equilateral triangle. Divide each side in half and remove the middle triangle. Repeat this process indefinitely. 9.8

Sierpinski Gasket What happens to the perimeter as you do more iterations? What about area? What is the fractal dimension of the gasket? Does this make sense? 9.8

Koch Snowflake Start with equilateral triangle. Iteration rule: 9.8 What happens to the perimeter? Area? What is the fractal dimension?