pine cone Ratio = 13:8 or 8:5

Transcription

1 Chapter 10: Introducing Geometry 10.1 Basic Ideas of Geometry Geometry is everywhere o Road signs o Carpentry o Architecture o Interior design o Advertising o Art o Science Understanding and appreciating geometry is basic to understanding and appreciating mathematics Focus for this chapter is on visualizing o Two-dimensional figures and their properties o Three-dimensional figures and their properties o Further development of your spatial sense Seeing Geometry in the World Geometry in nature honeycombs snowflakes Fibonacci sequence , 1, 2, 3, 5, 8, 13, 21, 34, 55, sunflowers Ratio of counterclockwise spirals to clockwise spirals is often 55:34 or 34:21

2 pine cone Ratio = 13:8 or 8: Golden ratio Approximately Ratio of successive Fibonacci numbers star fish snail shell Geometry in human endeavors geometry = earth measure Egyptian pyramids Pentagon Modeling and Defining Basic Geometric Ideas Points, lines, planes, and space point no length, no width, no height space the set of all points that has no boundaries line points in a straight, unlimited length with no width, height, or endpoints Two different points define exactly one line plane set of all points on a flat surface with no height and no edges

3 Three points not contained in exactly one line define exactly one plane See figures p collinear points contained on one line coplanar points contained in one plane intersect lines two distinct lines intersect in exactly one point called the point of intersection planes two distinct planes intersect in exactly one line called the line of intersection parallel two distinct lines in the same plane that do not intersect skew two distinct lines in two different planes that do not intersect Segments, rays, angles See table 10.1 p line segment set of points A and B ad all of the points between A and B ray point A and all of the points on AB on the same side of A as point B angle the union of two rays with a common endpoint length measure assignment of a real number of some unit to a segment congruent line segments have the same length midpoint of a segment M is the midpoint of EF if and only if EM MF bisector of the segment any point, line segment, ray, line or plane that contains the midpoint of the segment degree measure real number between 0 and 360 degrees that defines the amount of rotation or size of an angle protractor a device for measuring angles straight angle 180 angle reflex angle > 180, but < zero angle - 0 or no rotation interior points inside angle

4 exterior points outside angle angle bisector an interior ray that divides the measure of the angle into two congruent angles Special angles and perpendicular lines See table 10.2 p right angle acute angle 0 < angle < obtuse angle 90 < angle < complementary angles sum of two angles is supplementary angles sum of two angles is adjacent two angles with same vertex and a common side, with no common interior points linear pair pair of adjacent angles with two non-common sides on the same line, also form supplementary angles vertical angles pair of angles formed by two intersecting lines and that are not a linear pair of angles perpendicular two lines are perpendicular if and only if they intersect to form four right angles perpendicular bisector of a segment is a perpendicular line which passes through the midpoint of the line segment Circles and polygons open curve path in a plane with different starting and ending points closed curve path in a plane with the same starting and ending point non-simple closed curve closed curve that has more points in common than the starting and ending points simple closed curve closed curve that only has the beginning and ending points in common circle special simple closed curve where all points in the curve are equidistant from a given point in the same plane center middle point of the circle chord line segment connecting two distinct points on the circle diameter is a chord that passes through the center of the circle radius line segment connecting the center of the circle to any point on the circle polygon closed curve created by the union of line segments meeting at their endpoints such that at most, two segments meet at one point each segment meets exactly two other segments at their endpoints non-simple polygon at least one pair of line segments intersect in a point other than their endpoints simple polygon follows polygon rules simple convex polygon line test: draw a line through the polygon; No line can be drawn such that the line is intersected by the polygon more than twice

5 simple non-convex (concave) polygon - line test: draw a line through the polygon; At least one line can be drawn such that the line is intersected by the polygon more than twice n-gon the whole number n represents the number of sides for the polygon: a triangle is a 3-gon; a square is a 4-gon interior angle formed by two sides of the polygon with a common vertex regular polygon simple polygon where the all the line segments and all of the angles are congruent See table 10.3 p Triangles Union of three line segments formed by three distinct non-collinear points vertices intersection points of line segments forming the angles of the polygon sides the line segments forming the polygon median line segment connecting a vertex to the midpoint of the side opposite the vertex altitude line segment from a vertex of a triangle to a line containing the side of the triangle opposite the vertex See table 10.4 p Quadrilaterals Square and a rectangle are special types of kites? See table 10.5 p Problems and Exercises p Home work: 1, 9-15, 23, 27, 35, 57, 64