14365C06.pgs 7/12/07 2:57 PM Page 209 CHAPTER 6 TRANSFORMATIONS AND THE COORDINATE CHAPTER PLANE TABLE OF CONTENTS 6-1 The Coordinates of a Point in a Plane In our study of mathematics, we are accustomed to representing an equation as a line or a curve. This 6-2 Line Reflections blending of geometry and algebra was not always famil- 6-3 Line Reflections in the Coordinate Plane iar to mathematicians. In the seventeenth century, René Descartes (1596–1650), a French philosopher 6-4 Point Reflections in the Coordinate Plane and mathematician, applied algebraic principles and 6-5 Translations in the methods to geometry. This blending of algebra and Coordinate Plane geometry is known as coordinate geometry or 6-6 Rotations in the Coordinate analytic geometry. Pierre de Fermat (1601–1665) Plane independently developed analytic geometry before 6-7 Glide Reflections Descartes but Descartes was the first to publish his 6-8 Dilations in the Coordinate work. Plane Descartes showed that a curve in a plane could be 6-9 Transformations as Functions completely determined if the distances of its points Chapter Summary from two fixed perpendicular lines were known. We Vocabulary call these distances the Cartesian coordinates of the points; thus giving Descartes’ name to the system Review Exercises that he developed. Cumulative Review 209 14365C06.pgs 7/12/07 2:57 PM Page 210 210 Transformations and the Coordinate Plane 6-1 THE COORDINATES OF A POINT IN A PLANE In this chapter, we will review what you know about the coordinate plane and y you will study transformations and symmetry, which are common elements in 3 nature, art, and architecture. 2 1 Two intersecting lines determine a plane. The coordinate plane is deter- O mined by a horizontal line, the x-axis, and a vertical line, the y-axis, which are –3 –2 –1 1 2 3 x –1 perpendicular and intersect at a point called the origin. Every point on a plane –2 can be described by two numbers, called the coordinates of the point, usually –3 written as an ordered pair. The first number in the pair, called the x-coordinate or the abscissa, is the distance from the point to the y-axis. The second number, the y-coordinate or the ordinate is the distance from the point to the x-axis. In general, the coordinates of a point are represented as (x, y). Point O, the origin, has the coordinates (0, 0). We will accept the following postulates of the coordinate plane. Postulate 6.1 Two points are on the same horizontal line if and only if they have the same y-coordinates. Postulate 6.2 The length of a horizontal line segment is the absolute value of the differ- ence of the x-coordinates. Postulate 6.3 Two points are on the same vertical line if and only if they have the same x-coordinate. Postulate 6.4 The length of a vertical line segment is the absolute value of the difference of the y-coordinates. Postulate 6.5 Each vertical line is perpendicular to each horizontal line. Locating a Point in the Coordinate Plane An ordered pair of signed numbers uniquely determines the location of a point in the plane. 14365C06.pgs 7/12/07 2:57 PM Page 211 The Coordinates of a Point in a Plane 211 Procedure To locate a point in the coordinate plane: 1. From the origin, move along the x-axis the number of units given by the x- coordinate. Move to the right if the number is positive or to the left if the number is negative. If the x-coordinate is 0, there is no movement along the x-axis. 2. Then, from the point on the x-axis, move parallel to the y-axis the number of units given by the y-coordinate. Move up if the number is positive or down if the number is negative. If the y-coordinate is 0, there is no move- ment in the y direction. y O x For example, to locate the point A(Ϫ3, Ϫ4), from O, move 3 units to the left along the x-axis, then 4 units down, parallel to the y-axis. A Finding the Coordinates of a Point in a Plane The location of a point in the coordinate plane uniquely determines the coordi- nates of the point. Procedure To find the coordinates of a point: 1. From the point, move along a vertical line to the x-axis.The number on the x-axis is the x-coordinate of the point. 2. From the point, move along a horizontal line to the y-axis.The number on the y-axis is the y-coordinate of the point. y O For example, from point R, move in the verti- 123456 x cal direction to 5 on the x-axis and in the hori- zontal direction to Ϫ6 on the y-axis. The coordinates of R are (5, Ϫ6). Note: The coordinates of a point are often referred to as rectangular coordinates. R(5, –6) Ϫ7 14365C06.pgs 7/12/07 2:57 PM Page 212 212 Transformations and the Coordinate Plane Graphing Polygons A quadrilateral (a four-sided polygon) can be represented in the coordinate plane by locating its vertices and then drawing the sides connecting the vertices in order. The graph shows the quadrilateral ABCD. y The vertices are A(3, 2), B(Ϫ3, 2), C(Ϫ3, Ϫ2) and D(3, Ϫ2). A(3, 2) 1. Points A and B have the same y-coordi- 1 nate and are on the same horizontal line. O 2. Points C and D have the same y-coordi- 1 x nate and are on the same horizontal line. 3. Points B and C have the same x-coordi- nate and are on the same vertical line. 4. Points A and D have the same x-coordi- nate and are on the same vertical line. 5. Every vertical line is perpendicular to every horizontal line. 6. Perpendicular lines are lines that intersect to form right angles. Each angle of the quadrilateral is a right angle: mЄA ϭ mЄB ϭ mЄC ϭ mЄD ϭ 90 From the graph, we can find the dimensions of this quadrilateral.To find AB and CD, we can count the number of units from A to B or from C to D. AB ϭ CD ϭ 6 Points on the same horizontal line have the same y-coordinate. Therefore, we can also find AB and CD by subtracting their x-coordinates. AB ϭ CD ϭ 3 Ϫ (Ϫ3) ϭ 3 ϩ 3 ϭ 6 To find BC and DA, we can count the number of units from B to C or from D to A. BC ϭ DA ϭ 4 Points on the same vertical line have the same x-coordinate. Therefore, we can find BC and DA by subtracting their y-coordinates. BC ϭ DA ϭ 2 Ϫ (Ϫ2) ϭ 2 ϩ 2 ϭ 4 14365C06.pgs 7/12/07 2:57 PM Page 213 The Coordinates of a Point in a Plane 213 EXAMPLE 1 Graph the following points: A(4, 1), B(1, 5), C(Ϫ2,1). Then draw ᭝ABC and find its area. Solution The graph shows ᭝ABC. y B(1, 5) To find the area of the triangle, we need to know the lengths of the base and of the alti- tude drawn to that base. The base of ᭝ABC is AC, a horizontal line segment. AC ϭ 4 Ϫ (Ϫ2) O D(1, 1) A(4, 1) ϭ 4 ϩ 2 1 x ϭ 6 The vertical line segment drawn from B perpendicular to AC is the altitude BD. BD ϭ 5 Ϫ 1 ϭ 4 ϭ 1 Area 2(AC)(BD) ϭ 1 2(6)(4) ϭ 12 Answer The area of ᭝ABC is 12 square units. Exercises Writing About Mathematics 1. Mark is drawing isosceles right triangle ABC on the coordinate plane. He locates points A(Ϫ2, Ϫ4) and C(5, Ϫ4). He wants the right angle to be /C. What must be the coordinates of point B? Explain how you found your answer. 2. Phyllis graphed the points D(3, 0), E(0, 5), F(Ϫ2, 0), and G(0, Ϫ4) on the coordinate plane and joined the points in order. Explain how Phyllis can find the area of this polygon. Developing Skills In 3–12: a. Graph the points and connect them with straight lines in order, forming a polygon. b. Find the area of the polygon. 3. A(1, 1), B(8, 1), C(1, 5) 4. P(0, 0), Q(5, 0), R(5, 4), S(0, 4) 5. C(8, Ϫ1), A(9, 3), L(4, 3), F(3, Ϫ1) 6. H(Ϫ4, 0), O(0, 0), M(0, 4), E(Ϫ4, 4) 14365C06.pgs 7/12/07 2:57 PM Page 214 214 Transformations and the Coordinate Plane 7. H(5, Ϫ3), E(5, 3), N(Ϫ2, 0) 8. F(5, 1), A(5, 5), R(0, 5), M(Ϫ2, 1) 9. B(Ϫ3, Ϫ2), A(2, Ϫ2), R(2, 2), N(Ϫ3, 2) 10. P(Ϫ3, 0), O(0, 0), N(2, 2), D(Ϫ1, 2) 11. R(Ϫ4, 2), A(0, 2), M(0, 7) 12. M(Ϫ1, Ϫ1), I(3, Ϫ1), L(3, 3), K(Ϫ1, 3) 13. Graph points A(1, 1), B(5, 1), and C(5, 4). What must be the coordinates of point D if ABCD is a quadrilateral with four right angles? 14.