AMSCO'S Geometry. New York - Rye High School

Line Reflections in the Coordinate Plane 223(2) The y-axis bisects PPr. Let Q be the point at which PPr intersects they-axis. The x-coordinate of every point on the y-axis is 0. The length of ahorizontal line segment is the absolute value of the difference of thex-coordinates of the endpoints.PQ a 0 a and PQ a 0 = aSince PQ PQ, Q is the midpoint of PPr or the y-axis bisects PPr.Steps 1 and 2 prove that if P has the coordinates (a, b) and P has the coordinates(a, b), the y-axis is the perpendicular bisector of PPr, and therefore,the image of P(a, b) is P(a, b).Reflection in the x-axisIn the figure, ABC is reflected in the x-axis. Itsimage under the reflection is ABC. Fromthe figure, we see that:A(1, 2) → A(1, 2)B(3, 4) → B(3, 4)C(1, 5) → C(1, 5)For each point and its image under a reflectionin the x-axis, the x-coordinate of the imageis the same as the x-coordinate of the point; they-coordinate of the image is the opposite of they-coordinate of the point. Note that for a reflectionin the x-axis, the image of (1, 2) is (1, 2)and the image of (1, 2) is (1, 2).1OA(1, 2)1xA reflection in the x-axis can be designated as r x-axis. For example, if theimage of (1, 2) is (1, 2) under a reflection in the x-axis, we can write:r x-axis(1, 2) (1, 2)From these examples, we form a general rule that can be proven as a theorem.yC(1, 5)B(3, 4)Theorem 6.3Under a reflection in the x-axis, the image of P(a, b) is P(a, b).The proof follows the same general pattern asthat for a reflection in the y-axis. Prove that thex-axis is the perpendicular bisector of PPr.The proofis left to the student. (See exercise 18.)yOP(a, b)xQ(a, 0)P(a, b)