8/22/2023 0 Comments Translation reflection rotation![]() ![]() ∴ A"B" is the image of AB under an enlargement E(O, kk') with centre O and scale factor kk'. Let A"B" be the image of A'B' under the enlargement E 2 with centre O and scale factor k'. Let E 2 be the enlargement with centre O and scale factor k'. Let A'B' be the image of the given line AB under the enlargement E 1 with centre o and the scale factor k. Let E 1(O, k) be the enlargement with centre O and the scale factor k. There are two cases in the combination of two enlargements: 1. Similarly, an object once reduced can further be reduced. Combination of Two EnlargementsĪn object once enlarged can further be enlarged. ![]() In fact ΔP"Q"R" is the image of ΔPQR under a translation whose magnitude is twice the distance between AB and CD. ΔP'Q'R' is the image of Δpqr under the reflection in the line AB and ΔP"Q"R" is the image of ΔP'Q'R'. In the following figure, AB and CD are two parallel straight lines. In fact, ΔA"B"C" is the image of ΔABC under the rotation about O through an angle 2∠POQ.įig:- combination of two reflections over two parallel lines ΔA'B'C' is the image of ΔABC under the reflection in the line OP and ΔA"B"C" is the image of ΔA'B'C' under the reflection in the line OQ. In the following figure, O is the point of intersection of two straight lines OP and OQ. ![]() the direction from P to P").Ī reflection followed by a reflection is equivalent to either a translation or a rotation.įig:- combination of two reflections over intersecting lines The direction of rotation is the direction from OA to OB (i.e. Hence, if the axes of reflections intersect at a point O, then a reflection followed by another reflection is equivalent to a rotation about the centre O through the angle twice the angle between the axes of reflection. Here, P" be the image under the reflection in the line OA followed by the reflection in the line OB. Then, OB is the perpendicular bisector of P'P". Let P" is the image of P' under the reflection in the line OB. Then, OB is the perpendicular bisector of PP'. Let P" be the image of P' under the reflection in the line OB. Let OA and OB intersect each other at the point O. When the axes of reflection intersect at a point When the axes of reflection intersect at a point The distance of the translation is twice the distance between the axes of reflections and the direction is perpendicular to the axis of reflections. Hence, if the axis of reflection is parallel, a reflection followed by another reflection is equivalent to the translation. The composite transformation of R with itself is denoted by R.R = R 2 Combination of Two Translations Combination of two translationsĪ translation followed by a translation is equivalent to a single translation. If T and W are two different transformations, then the product TW gives the combination of the transformations W followed by T (this means to work out W first and then apply T), which can be denoted by single transformation say Z and we write it as Z = TW. R 1R 2 gives a combination of transformations R 2 followed by R 1 whereas, R 2R 1 gives the combination of transformations R 1 followed by R 2. The transformations R 1R 2 and R 2R 1 have different meanings. It is denoted by R 2 or R 1.The composite transformation R 2 R 1 is also called as the transformation R 1 followed byR 2. Then, the transformation which maps a point P to P'' is said to be the combination of R 1 and R 2 or it is said to be the composite transformation of R 1 and R 2. Let R 1 be a transformation which maps a point P to the point P' and R 2 be another transformation which maps P' to the point p''. After a combination of transformations, the change from the single object to the final image can be described by a single transformation. Such transformation is called a combination of transformations. When an object has been transformed, its image can again be transformed to form a new image. The transformations discussed above are the examples of single transformation. ![]()
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