# If a point other than the origin happens to lie on

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in Quadrant IV. If a point other than the origin happens to lie on the axes, we typically refer tothat point as lying on the positive or negative x-axis (if y = 0) or on the positive or negative y-axis(if x = 0). For example, (0; 4) lies on the positive y-axis whereas (?117; 0) lies on thenegativex-axis. Such points do not belong to any of the four quadrants.One of the most important concepts in all of Mathematics is symmetry.11 There are many types ofsymmetry in Mathematics, but three of them can be discussed easily using Cartesian Coordinates.De_nition 1.3. Two points (a; b) and (c; d) in the plane are said to be• symmetric about the x-axis if a = c and b = ?d• symmetric about the y-axis if a = ?c and b = d• symmetric about the origin if a = ?c and b = ?dSchematically,0 xyQ(?x; y) P(x; y)R(?x;?y) S(x;?y)In the above _gure, P and S are symmetric about the x-axis, as are Q and R; P and Qaresymmetric about the y-axis, as are R and S; and P and R are symmetric about the origin, as areQ and S.Example 1.1.3. Let P be the point (?2; 3). Find the points which are symmetric to P about the:1. x-axis 2. y-axis 3. originCheck your answer by plotting the points.
11According to Carl. Je_ thinks symmetry is overrated.10 Relations and Functions1. To _nd the point symmetric about the x-axis, we replace the y-coordinate with its oppositeto get (?2;?
3. To _nd the point symmetric about the origin, we replace the x- and y-coordinates with theiropposites to get (2;?3).xyP(?2; 3)(?2;3)(2; 3)(2;3)?3 ?2 ?1 1 2 3?3?2?1123One way to visualize the processes in the previous example is with the concept of a reection. Ifwe start with our point (?2; 3) and pretend that the x-axis is a mirror, then the reection of (?2; 3)across the x-axis would lie at (?2;?3). If we pretend that the y-axis is a mirror, the reectionof (?2; 3) across that axis would be (2; 3). If we reect across the x-axis and then the y-axis, wewould go from (?2; 3) to (?2;?3) then to (2;?3), and so we would end up at the point symmetricto (?2; 3) about the origin. We summarize and generalize this process below.ReectionsTo reect a point (x; y) about the:• x-axis, replace y with ?y.• y-axis, replace x with ?x.• origin, replace x with ?x and y with ?1.1.3 Distance in the PlaneAnother important concept in Geometry is the notion of length. If we are going to unite Algebraand Geometry using the Cartesian Plane, then we need to develop an algebraic understanding ofwhat distance in the plane means. Suppose we have two points, P (x0; y0) and Q(x1; y1) ; in theplane. By the distance d between P and Q, we mean the length of the line segment joining P withQ. (Remember, given any two distinct points in the plane, there is a unique line containing both1.1 Sets of Real Numbers and The Cartesian Coordinate Plane 11points.) Our goal now is to create an algebraic formula to compute the distance between these twopoints. Consider the generic situation below on the left.P (x0; y0)Q(x1; y1)dP (x0; y0)Q(x1; y1)dy.
(x1; y0)