Math 366 Definitions and Theorems

Transcription

1 Math 366 Definitions and Theorems Chapter 11 In geometry, a line has no thickness, and it extends forever in two directions. It is determined by two points. Collinear points are points on the same line. (Any two points are collinear but not every three points have to be collinear.) A point C is between points A and B if C A, C B, and C is on the part of the line flanked by A and B. Alternate Definition: Point B is between points A and C if A, B, and C are collinear and the sum of the distances from A to B and B to C equals the distance from A to C. A line segment is a subset of a line that contains two points of the line and all points between those two points. Notation: AB or BA. A ray is a subset of a line that contains the endpoint and all points on the line on one side of the point. Notation: AB A plane has no thickness, and it extends indefinitely in two directions. A plane is determined by three points that are not all on the same line. In other words, given three noncollinear points, a unique plane is determined. Points in the same plane are coplanar. Noncoplanar points cannot be placed in a single plane. Lines in the same plane are coplanar lines. Skew lines are lines that do not intersect, and there is no plane that contains them. Intersecting lines are two coplanar lines with exactly one point in common. Concurrent lines are lines that contain the same point. Two distinct coplanar line m and n that have no points in common are parallel lines. Four Axioms of Euclid There is exactly one line that contains any two distinct points. If two points lie in a plane, then the line containing the points lies in the plane. 1

2 If two distinct planes intersect, then their intersection is a line. There is exactly one plane that contains any three distinct noncollinear points. Three Theorems of Euclid A line and a point not on the line determine a plane. Two parallel lines determine a plane. Two intersecting lines determine a plane. Two distinct planes either intersect in a line or are parallel. (see p. 685) A line and a plane can be related in one of three ways: If a line and a plane have no point in common, the line is parallel to the plane. If two points of a line are in the plane, then the entire line containing the points is contained in the plane. If a line intersects the plane but is not contained in the plane, it intersects the plane at only one point. If a line is contained in a plane, it separates the plane into two half-planes. The plane is the union of three disjoint sets: the two half-planes, and the line that separates them. When two rays share an endpoint, an angle is formed. The rays of an angle are the sides of the angle, and the common endpoint is the vertex of the angle. Notation: ABC, CBA, B. 2

3 Adjacent angles share a common vertex and a common side and do not have overlapping interiors. An angle is measured according to the amount of opening between its sides. The degree is commonly used to measure angles. A complete rotation about a point has a measure of degrees, written 360. One degree is of a complete rotation. 360 A degree is subdivided into 60 equal parts, minutes, and each minute is divided into 60 equal parts, seconds. The measuring device pictured is a protractor. The degree is not the only unit used to measure angles. Radians are sometimes used in calculus and science; its measure is the angle whose arc length is the same as the radius (approximately 1 57 ). Grads are sometimes used in civil engineering; its measure is of 90, or An acute angle has measure less than 90. A right angle has measure 90. An obtuse angle has measure between 90 and 180. A straight angle has measure 180. In higher mathematics and in scientific applications, it is important to view an angle as being created by a ray rotating about its endpoint. If the ray makes one full rotation, we say that is sweeps an angle of 360. Angles with positive measure are created by a counterclockwise rotation; angles with negative measure by a clockwise rotation. Angles whose measure is greater than 360 are created when the ray makes more than one full rotation. When two lines intersect so that the angles formed are right angles, the lines are perpendicular lines. Two intersecting segments and/or rays are perpendicular if they lie on perpendicular lines. Notation: m n, AB AC, etc. A line perpendicular to a plane is a line that is perpendicular to every line in the plane through its intersection with the plane. 3

4 Draw a path on a piece of paper without lifting the pencil and without retracing any part of the path except single points. The drawing is called a curve. (see p. 698). A simple curve is one that does not intersect itself. A closed curve is one with no endpoints and which completely encloses an area. A convex curve is a simple, closed curve in which any straight line that crosses the curve crosses it at just two points. A concave curve is one that is not convex. A polygon is a simple, closed curve made by joining line segments, where each line segment intersects exactly two others. Notation: ABCD, BCDA, etc. (where A, B, C, and D are consecutive vertices). An n-gon is a polygon with n sides. Number of Sides Name 3 Triangle 4 Quadrilateral 5 Pentagon 6 Hexagon 7 Heptagon 8 Octagon 9 Nonagon 10 Decagon A polygon and its interior make up a polygonal region. Any two sides of a polygon having a common vertex determine an interior angle or angle of the polygon. An exterior angle of a convex polygon is determined by a side of the polygon and the extension of a contiguous side of the polygon. A diagonal is a line segment connecting nonconsecutive vertices of a polygon. Congruent parts are parts with the same size and shape. Congruent segments can be fitted exactly one top of each other. (They have the same length.) Notation: AB CD ( AB CD iff AB = CD) Congruent angles have the same measure. ( A B iff m A = m B) A regular polygon is one in which all the interior angles are congruent and all the sides are congruent. A regular polygon is both equiangular and equilateral. A triangle containing one right angle is a right triangle. A triangle in which all the angles are acute is an acute triangle. A triangle containing one obtuse angle is an obtuse triangle. 4

5 A triangle with no congruent sides is a scalene triangle. A triangle with at least two congruent sides is an isosceles triangle. A triangle with three congruent sides is an equilateral triangle. A trapezoid is a quadrilateral with at least one pair of parallel sides. (Some texts define a trapezoid as a quadrilateral with exactly one pair of parallel sides.) A kite is a quadrilateral with two adjacent sides congruent and the other two sides also congruent. An isosceles trapezoid is a trapezoid with exactly one pair of congruent sides. (Equivalently, an isosceles trapezoid is a trapezoid with two congruent base angles.) A parallelogram is a quadrilateral in which each pair of opposite sides is parallel. A rectangle is a parallelogram with a right angle. (Equivalents, a rectangle is a quadrilateral with four right angles.) A rhombus is a parallelogram with two adjacent sides congruent. (Equivalently, a rhombus is a quadrilateral with all sides congruent.) A square is a rectangle with two adjacent sides congruent. (Equivalently, a square is a quadrilateral with four right angles and four congruent sides.) Vertical angles are pairs of angles opposite each other at the intersection of two lines. Theorem 11-1 Vertical angles are congruent. Supplementary angles are two angles, the sum of whose measures is 180. Each angle is a supplement of the other. Complementary angles are two angles, the sum of whose measures is 90. Each angle is a complement of the other. A line that intersects a pair of lines in a plane is a transversal of those lines. If any two distinct coplanar lines are cut by a transversal, then a pair of corresponding angles, alternate interior angles, or alternate exterior angles are congruent if and only if the lines are parallel. Corollary : If any two distinct coplanar lines are cut by a transversal, then a pair of same side interior angles are supplementary iff the lines are parallel. Theorem 11-2 The sum of the measures of the interior angles of a triangle is 180. Theorem 11-3 The sum of the measures of the exterior angles (one at each vertex) of a convex polygon is

6 Theorem 11-4 a. The sum of the measures of the interior angles of any convex polygon with n sides is 180n 360, or (n 2)180. b. The degree measure of a single interior angle of a regular n-gon is ( n 2) 180. n A simple closed surface has exactly one interior, no holes, and is hollow n 360, or n A sphere is the set of all points at a given distance from a given point, the center. A sphere is a simple closed surface. A solid is a simple closed surface with all interior points. A polyhedron is a simple closed surface made up of polygonal regions, or faces. The vertices of the polygonal regions are the vertices of the polyhedron, and the sides of each polygonal region are the edges of the polyhedron. A prism is a polyhedron in which two congruent faces lie in parallel planes and the other faces are bounded by parallelograms. The parallel faces of a prism are the bases of the prism. A prism is usually names after its bases. The faces other than the bases are the lateral faces of a prism. A right prism is one in which the lateral faces are all bounded by rectangles. An oblique prism is one in which some of the lateral faces are not bounded by rectangles. A pyramid is a polyhedron determined by a polygon and a point not in the plane of the polygon. The pyramid consists of the triangular regions determined by the point and each pair of consecutive vertices of the polygon and the polygonal region determined by the polygon. The polygonal region is the base of the pyramid, and the point is the apex. The faces other than the base are lateral faces. Pyramids are classified according to their bases. A right pyramid is one in which all the lateral faces are congruent isosceles triangles. A polyhedron is a convex polyhedron if and only if the segment connecting any two points in the interior of the polyhedron is itself in the interior. A regular polyhedron is a convex polyhedron a) whose faces are congruent regular polygonal regions, and b) the number of edges that meet at each vertex is the same for all the vertices of the polyhedron. A cylinder is the surface formed by a line segment and a line when the segment moves to that it always remains parallel to the line, so that the endpoints trace a simple closed planar curve other than a polygon, the simple closed curves, and their interiors. The simple closed curves traced by the endpoints of the segment, along with their interiors, are the bases of the cylinder; the remaining points constitute the lateral surface of the cylinder.

7 A circular cylinder is one whose base is a circular region. A right cylinder is one in which the line segment forming the cylinder is perpendicular to a base. An oblique cylinder is any cylinder that is not a right cylinder. A cone is the union of line segments connecting a point to each point of a simple closed curve, not coplanar with the point, the simple closed curve, and the interior of the curve. The point is the vertex of the cone; the points not in the base constitute the lateral surface of the cone. A line segment from the vertex perpendicular to the plane of the base is the altitude of the cone. A right circular cone is one whose altitude intersects the circular base at the center of the circle. An oblique circular cone is a circular cone that is not a right circular cone. The problem can be made simpler by representing the problem in a network as shown below: The points in the network are vertices and the curves are arcs. A network that has a path such that each arc is passed through exactly once is traversable. A network that is traversable in such a way that the starting point and the ending point are the same is an Euler circuit. An even vertex is one for which the number of arcs meeting there is even. An odd vertex is one for which the number of arcs meeting there is odd. Properties of a Network: In general, networks have the following properties: 1) If a network has all even vertices, it is traversable. Any vertex can be a starting point, and the same vertex must be the stopping point. Thus, the network is an Euler circuit. 2) If a network has two odd vertices, it is traversable. One odd vertex must be the starting point, and the other odd vertex must be the stopping point. 7

8 Chapter 12 Similar objects have the same shape but not necessarily the same size. Notation: ABC DEF Congruent objects have the same shape and same size. Notation: ABC DEF A circle is the set of all points in a plane equidistant from a given point, its center. The distance is the radius. The word radius is used to describe both the segment from the center to a point on a circle and the length of that segment. Definition of congruent segments: AB CD iff AB = CD Definition of congruent angles: ABC DEF iff m ABC = m DEF. Geometric constructions are done with only a writing instrument, straight edge, and compass. Definition of congruent triangles: ABC DEF iff A D, B E, C F, CPCTC Corresponding parts of congruent triangles are congruent. AB DE, BC EF, and AC DF. Theorem 12-1 Side, Side, Side (SSS) If the three sides of one triangle are congruent, respectively, to the three sides of a second triangle, then the triangles are congruent. Theorem 12-2 Triangle Inequality The sum of the measures of any two sides of a triangle must be greater than the measure of the third side. Axiom 12-1 Side, Angle, Side (SAS) If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, respectively, then the two triangles are congruent. The perpendicular bisector of a segment is a segment, ray, or line that is perpendicular to the segment at its midpoint. An altitude of a triangle is the perpendicular segment from a vertex of the triangle to the line containing the opposite side of the triangle. Theorem 12-3 Hypotenuse-Leg (HL) If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent. 8

9 Theorem 12-4 a. Any point equidistant from the endpoints of a segment is on the perpendicular bisector of the segment. b. Any point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment. Theorem 12-5 For every isosceles triangle: a. The angles opposite to the congruent sides are congruent. (Base angles of an isosceles triangle are congruent.) b. The angle bisector of an angle formed by two congruent sides contains an altitude of the triangle and is the perpendicular bisector of the third side of the triangle. A circle is circumscribed about a triangle when all three vertices of the triangle are on the circle. The circle is called a circumcircle. Its center is called the circumcenter and its radius the circumradius. A line is tangent to a circle if it intersects the circle in one and only one point and is perpendicular to a radius. A circle is inscribed in a triangle if all the sides of the triangle are tangent to the circle. The inscribed circle is the incircle; the center is the incenter. Theorem 12-6 Angle, Side, Angle (ASA) If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, respectively then the triangles are congruent. Theorem 12-7 Angle, Angle, Side (AAS) If two angles and a corresponding side of one triangle are congruent to two angles and a corresponding side of another triangle, respectively, then the two triangles are congruent. A median is a segment connecting a vertex to the midpoint of the opposite side of a triangle. 9

10 Properties of Quadrilaterals Quadrilateral and Its Definition D A Trapezoid: A quadrilateral with at least one pair of parallel sides A D Parallelogram: A quadrilateral in which each pair of opposite sides is parallel D A Rectangle: A parallelogram with a right angle. A D B Kite: A quadrilateral with two distinct pairs of congruent adjacent sides A D B Rhombus: A parallelogram with all sides congruent. D C C C B C B B C C Properties of the Quadrilateral Consecutive angles between parallel sides are supplementary. a. A parallelogram has all the properties of a trapezoid. b. Opposite sides are congruent. c. Opposite angles are congruent. d. Diagonals bisect each other. a. A rectangle has all the properties of a parallelogram. b. All the angles of a rectangle are right angles. c. A quadrilateral in which all the angles are right angles is a rectangle. d. The diagonals of a rectangle are congruent and bisect each other. a. Lines containing the diagonals are perpendicular to each other. b. A line containing one diagonal is a bisector of the other. c. One diagonal bisects nonconsecutive angles. a. A rhombus has all the properties of a parallelogram and a kite. b. A quadrilateral in which all the sides are congruent is a rhombus. c. The diagonals of a rhombus are perpendicular to and bisect each other. d. Each diagonal bisects opposite angles. A square has all the properties of a parallelogram, a rectangle, and a rhombus. A B Square: A rectangle with all sides congruent 10

11 A rhombus is a parallelogram in which all the sides are congruent. An angle bisector is a ray that separates an angle into two congruent angles. The distance from a point to a line is the length of the perpendicular segment from the point to the line. Theorem 12-8 a. Any point P on an angle bisector is equidistance from the sides of the angle. b. Any point that is equidistant from the sides of an angle is on the angle bisector of the angle. Theorem 12-9 The angle bisectors of a triangle are concurrent and the three distances from the point of intersection to the sides are equal. A line is tangent to a circle if it intersects the circle in one and only one point and is perpendicular to a radius. A circle is inscribed in a triangle if all the sides of the triangle are tangent to the circle. The inscribed circle is the incircle; the center is the incenter. Two figures that have the same shape but not necessarily the same size are similar. The ratio of the corresponding side lengths is the scale factor. Theorem SSS Similarity for Triangles If corresponding sides of two triangles are proportional, then the triangles are similar. Theorem SAS Similarity for Triangles Given two triangles, if two sides are proportional and the included angles are congruent, then the triangles are similar. Theorem Angle, Angle (AA) Similarity for Triangles If two angles in one triangle are congruent, respectively, to two angles of a second triangle, then the triangles are similar. Theorem If a line parallel to one side of a triangle intersects the other sides, then it divides those sides into proportional segments. Theorem If a line divides two sides of a triangle into proportional segments, then the line is parallel to the third side. Theorem If parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on any transversal. 11

12 A midsegment is a segment that connects midpoints of two adjacent sides of a triangle or quadrilateral. Theorem The Midsegment Theorem The midsegment is parallel to the third side of the triangle and half as long. Theorem If a line bisects one side of a triangle and is parallel to a second side, then it bisects the third side and therefore is a midsegment. A Cartesian coordinate system is constructed by placing two number lines perpendicular to each other. The intersection point of the two lines is the origin, the horizontal lines is the x-axis; the vertical line is the y-axis. The location of any point can be described by an ordered pair of numbers. The first component in the ordered pair is the abscissa, or x-coordinate; the second component is the ordinate, or y-coordinate. The graph of a set is the resulting picture on the Cartesian coordinate system from a plot of all points that satisfy a given condition. The graph of the equation x = a, where a is some real number, is a line perpendicular to the x-axis through the points with coordinates (a, 0). The graph of the equation y = b is a line perpendicular to the y-axis through the point with coordinates (0, b). Equation of a Line Every line has an equation of the form either y = mx + b or x = a, where m is the slope and (0, b) is the y-intercept. Slope Formula Given two points A (x 1, y 1 ) and B(x 2, y 2 ) with x 1 x 2, the slope m of the line AB is rise y y2 y1 m= = = run x x x Point-Slope Formula y y 1 = m(x x 1 ) 2 1 The word trigonometry is derived from the Greek words tronom, which means triangle, and metron, which means measurement. The hypotenuse of a right triangle is the longest side and is opposite the right angle. Pythagorean Theorem: In a right triangle, with hypotenuse c and sides a and b, c 2 = a 2 + b 2. In a triangle, the length of the side opposite the 30 angle is half the length of the hypotenuse. 12

13 Trigonometric functions are defined as ratios of the lengths of the sides of a right triangle. sin( A) = cos( A) = tan( A) = a c b c a b length of opposite side = length of hypotenuse length of adjacent side = length of hypotenuse length of = length of opposite side adjacent side = = = opp hyp adj hyp opp adj 13

14 Chapter 12 The English System Originally, a yard was the distance from the tip of the nose to the end of an outstretched arm of an adult person and a foot was the length of a human foot. Unit yard (yd) foot (ft) mile (mi) Equivalent in Other Units 3 ft 12 in yd, or 5280 ft Dimensional analysis is a process of convert from one unit of measure to another, using unit ratios (ratios equivalent to 1). The metric system is a standard based on the meter. Prefix Unit Symbol Relationship to base unit kilo kilometer km 1000 m hector hectometer hm 100 m deka dekameter dam 10 m meter m base unit deci decimeter dm 0.1 m centi centimeter cm 0.01 m milli millimeter mm m Approximate Conversions Between English and Metric Systems 1 km 0.62 mi 1 m 1.09 yd 2.54 cm 1 in The greatest possible error (GPE) of a measurement is one-half the smallest unit used. Distance Properties 1. The distance between any two points A and B is greater than or equal to 0, written AB The distance between any two points A and B is the same as the distance between B and A, written AB = BA. 3. For any three points A, B, and C, the distance between A and B plus the distance between B and C is greater than or equal to the distance between A and C, written AB + BC AC. Triangle Inequality The sum of the lengths of any two sides of a triangle is greater than the length of the third side. The perimeter of a simple closed curve is the length of the curve. If a figure is a polygon, its perimeter is the sum of the lengths of its sides. A perimeter has linear measure. A circle is the set of all points in a plane that are equidistant from a given point, the center. The distance is the radius. 14

15 The circumference of a circle is the perimeter. The circumference of a circle is found by C = 2πr = πd, where r is the radius and d is the diameter. The arc length is a linear distance around a portion of the circumference of a circle. The length of an arc depends on the radius of the circle and the central angle determining the arc. Length is given in units (i.e., inches) Area is measured using square units and the area of a region is the number of nonoverlapping square units that covers the region. Area is given in square units (i.e., square inches or in 2 ) Volume is given in cubic units (i.e., cubic inches or in 3 ) Unit Symbol Relationship to Square Meter Square kilometer km 2 1,000,000 m 2 Square hectometer hm 2 10,000 m 2 Square decameter dam m 2 Square meter m 2 1 m 2 Square decimeter dm m 2 Square centimeter cm m 2 Square millimeter mm m 2 The common unit of land measure in the English system is the acre. Historically, an acre was the amount of land a man with one horse could plow in one day. Unit of Area Equivalent 1 a (are) 100 m 2 1 ha (hectare) 100 a or 10,000 m 2 or 1 hm 2 1 km 2 1,000,000 m 2 1 acre 4840 yd 2 1 mi acres 15

16 Area Formulas Figure Area variables Rectangle A = lw l = length; w = width Parallelogram A = bh b = base; h = height Triangle A = ½ bh b = base; h = height Trapezoid A = ½ (b 1 +b 2 )h b 1, b 2 = bases; h = height Kite A = ½ d 1 d 2 d 1, d 2, = diagonals Regular Polygon A = n( ½ sa) n = # sides; a = apothem; s = length of side Circle A = πr 2 r = radius Sector θ A = (πr 2 θ = angle; r = radius ) 360 Theorem 13-2 Pythagorean Theorem If a right triangle has legs of lengths a and b and hypotenuse of length c, then c 2 = a 2 + b 2. A triangle is an isosceles right triangle with legs of equal length and two 45 angles. Theorem 13-3: Triangle Relationships In an isosceles right triangle, if the length of each leg is a, then the hypotenuse has length a 2. Theorem 13-4: Triangle Relationships In a triangle, the length of the hypotenuse is two times as long as the leg opposite the 30 angle and the leg opposite the 60 is 3 times the shorter leg. Theorem 13-5: Converse of the Pythagorean Theorem If ABC is a triangle with sides of lengths a, b, and c such that a 2 + b 2 = c 2, then ABC is a right triangle with the right angles opposite the side of length c. Theorem 13-6: Distance Formula The distance between the points A(x 1, y 1 ) and B(x 2, y 2 ) is given by AB= ( x y x1) + ( y2 1) Theorem 13-7: Equation of a Circle with Center at the Origin An equation of a circle with the center at the origin and radius r is x 2 + y 2 = r 2. Theorem 13-8: Equation of a Circle with Center at (h, k) An equation of a circle with center (h, k) and radius r is (x h) 2 + (y k) 2 = r 2. The surface area is the sum of the areas of the faces (lateral and bases) of a three-dimensional object. The lateral surface area is the sum of the areas of the lateral faces. Nets are often used to help find the surface area. The surface area of a cube is S.A. = 6s 2, where s is the length of a side (edge). The surface area of a right prism with regular n-gon bases is S.A. = ph + 2B, where p is the perimeter of the base, h is the height of a lateral face, and B is the area of the base. 16

17 The surface area of a right circular cylinder is S.A. = 2πr 2 + 2πrh, where r is the radius of the circle, and h is the height of the cylinder. The surface area of a right regular pyramid is S.A. = ½ bln + B, where n is the number of faces, l is the slant height, b is the length of a side of the base, and B is the area of the base. The formula can be simplified to S.A. = ½ pl + B. The surface area for a right circular cone is S.A. = πr 2 + πrl, where r is the radius of the circle, and l is the slant height. A great circle of a sphere is a circle on the sphere whose radius is equal to the radius of the sphere. The surface area of a sphere is S.A. = 4πr 2. Surface area is the number of square units covering a three-dimensional figure; volume describes how much space a three-dimensional figure contains. The most commonly used metric units of volume are the cubic centimeter and the cubic meter. Unit Symbol Relationship to Liter Kiloliter kl 1000 L Hectoliter hl 100 L Dekaliter dal 10 L Liter L 1 L Deciliter dl 0.1 L Centiliter cl 0.01 L Milliliter ml L Basic units of volume in the English system are the cubic foot (1 ft 3 ), the cubic yard (1 yd 3 ), and the cubic inch (1 in 3.). In the United States, 1 gal = 231 in L 1 qt = ¼ gal. 58 in. 3 4 cups = 2 pints = 1 quart 3 teaspoons = 1 tablespoon 16 tablespoons = 8 ounces = 1 cup 1 L gallons 1.06 quart The volume of a rectangular prism, with area of the base B and height h is V = Bh. The volume V of a cylinder is the product of the area of the base B and the height h. If the base is a circle of radius r, then V = Bh = πr 2 h 1 The volume of a pyramid is V = Bh, where B is the area of the base and h is the height. 3 17

18 1 1 2 The volume of a circular cone is V = Bh= πr h, where B is the area of the base, h is the 3 3 height, and r is the radius of the circular base. Volume of a sphere: V π = π = (4 r ) r r. Mass is the quantity of matter. Weight is the force exerted by gravitational pull. On Earth, the terms are commonly interchanged. In the metric system, the fundamental unit for mass is the gram, denoted g. Unit Symbol Relationship to Gram Ton (metric) t 1,000,000 g Kilogram kg 1000 g Hectogram hg 100 g Dekagram dag 10 g Gram g 1 g Decigram dg 0.1 g Centigram cg 0.01 g Milligram Mg g For water, H 2 0, 1 gm = 1 cm 3 and 1 kg = 1 dm 3 = 1 L Temperature Freezing point of water Boiling point of water Celsius Fahrenheit Kelvin 0 C 32 F 273 K 100 C 212 F 373 K 18

19 Chapter 14 Any rigid motion that preserves length or distance is an isometry (meaning equal measure ). Any function from a plane to itself that is a one-to-one correspondence between a plane and itself is a transformation of the plane. A translation is a motion of a plane that moves every point of the plane a specified distance in a specified direction along a straight line. Properties of Translations A figure and its image are congruent. The image of a line is a line parallel to it. Theorem 14-1: Translation in a Coordinate System A translation is a function from the plane to the plane such that to every point (x, y) corresponds the point (x + a, y + b), where a and b are real numbers. Notation: (x, y) (x + a, y + b) A rotation or turn is a transformation of the plane determined by holding one point, the center, fixed and rotating the plane about this point by a certain amount in a certain direction (turn angle). Usually a positive rotation is counterclockwise, and a negative rotation is clockwise. A rotation of 360 about a point will move any point (and figure) onto itself. Such a transformation is an identity transformation. A rotation of 180 is a half-turn. Rotations are useful in determining turn symmetries. Theorem 14-2: Slopes of Perpendicular Lines Two lines, neither of which is vertical, are perpendicular iff their slopes m 1 and m 2 satisfy the condition m 1 m 2 = -1. Every vertical line has an undefined slope and is perpendicular to a line with slope 0. A reflection is an isometry in which a figure is reflected across a reflecting line, creating a mirror image. Unlike a translation or rotation, the reflection reverses the orientation of the original figure. The reflected figure is congruent to the original figure. A reflection in a line l is a transformation of a plane that pairs each point P of the plane with a point P in such a way that l is the perpendicular bisector of P P, as long as P is not on l. If P is on l, then P = P. A glide reflection is a transformation consisting of a translation followed by a reflection in a line parallel to the slide arrow. Two geometric figures are congruent iff one is an image of the other under a single isometry or under a composition of isometries. 19

20 When a ray of light bounces off a mirror or a billiard ball bounces off the rail of a billiards table, the angle of incidence, the angle formed by the incoming ray and a lie perpendicular to the mirror, is congruent to the angle of reflection, the angle between the reflected ray and the line perpendicular to the mirror. A size transformation from the plane to the plane with center O and scale factor r (r > 0) is a transformation that assigns to each point in the plane a point A such that O, A, and A are collinear and OA = r OA and so that O is not between A and A. Theorem 14-3: A size transformation with center O and scale factor r (r > 0) has the following properties: 1. The image of a line segment is a line segment parallel to the original segment and r times as long. 2. The image of an angle is an angle congruent to the original angle. Two figures are similar if it is possible to transform one onto the other by a sequence of isometries (translation, rotation, reflection) followed by a size transformation (dilation). A geometric figure has a line of symmetry l if it is its own image under a reflection in l. A figure has rotational symmetry, or turn symmetry, when the traced figure can be rotated less than 360 about some point P, the turn center, so that it matches the original figure. Any figure that has 180 rotational symmetry is said to have point symmetry about the turn center. Any figure with point symmetry is its own image under a half-turn. This makes the center of the half-turn the midpoint of a segment connecting a point and its image. A three-dimensional figure has a plane of symmetry when every point of the figure on one side of the plane has a mirror image on the other side of the plane. A three-dimensional figure has rotational symmetry when the figure can be rotated less than 360 about a line n, the axis of rotation, so that it matched the original figure. A tessellation of a plane (or space) is the filling of the plane (or space) with repetitions of figures in such a way that no figures overlap and there are no gaps. A regular tessellation is a tessellation made up of one type of regular polygon. When more than one type of regular polygon is used and the arrangement of the polygons at each vertex is the same the tessellation is semiregular. 20