Line $x = 2$ is to the right of $x = -2$. The fact that the translation here moves to the right comes from the fact that the Two reflections about $x = -2$ and $x = 2$ is a translation by $8$ unitsįurther experimentation with reflections about parallel lines will show that $A^\prime = (1,2)$ then $x = -2$ is the perpendicular bisector of $\overline$ we can translate by $8$ units to the right. This is on the line $y = 2$ which is perpendicular to the line $x = -2$. To see why, consider the point $A = (-5,2)$. $x = -2$ are not influenced by the reflection. Note that $C^\prime = C$ becuase $C$ lies on the line $x = -2$. A little more work in this direction shows the remarkable fact that all rigid motions of the plane can be written in terms of reflections.īelow is a picture of triangle $ABC$ and its reflection about $x = -2$, triangle Graph the image of the figure using the transformation given. Here again the order in which the reflections are performed will matter and the result will be a rotation. The teacher may also wish to examine the case of successive reflections about lines which are not parallel: for example the line $y = x$ and the $x$-axis. This is also a translation but is it the same translation as in part (c) of the problem? This is an important example of two transformations (reflections of the plane) which do not commute: that is, the order in which the reflections are implemented changes the outcome. If the teacher wishes to give students further work with reflections, a good question would be to have them first reflect $\triangle ABC$ about $x = 2$ and then about $x = -2$. Toolbar Image Use the polygon tool to construct the triangle ABC 4) Toolbar Image Use the reflection tool (9th tool over, under the reflection diagram) 5. If we denote by $L_1$ the first line, $x = -2$, and by $L_2$ the second line, $x = 2$, then the composition of reflection about $L_1$ followed by reflection about $L_2$ is a translation by twice the distance between $L_1$ and $L_2$ (in the direction going from $L_1$ to $L_2$). ![]() This task examines the composition of successive reflections about parallel lines. This approach is also valuable in preparation for the coordinate-free transformation approach to reflections which will be seen in high school. Another valid approach would use the geometry of the coordinate grid without making explicit reference to the coordinates of the triangle vertices. Students are not prompted in the question to list the coordinates of the different triangle vertices but this is a natural extension of the task. When reflecting shapes, a good understanding of symmetry and naming and plotting coordinates can be helpful.The goal of this task is to give students an opportunity to experiment with reflections of triangles on a coordinate grid. When reflecting shapes in non-vertical mirror lines, rotating the paper to make the mirror line vertical can help to visualise a problem. The new shape is congruent close congruent Shapes that are the same shape and size, they are identical. distance from the mirror line to its corresponding vertex on the image. on the original shape is the same perpendicular close perpendicular Perpendicular lines are at 90° (right angles) to each other. The line of reflection is also called the mirror line.Įach vertex close vertex The point at which two or more lines intersect (cross or overlap). A shape can be reflected across a line of reflection to create an image. is one of the four types of transformations close transformation A change in position or size, transformations include translations, reflections, rotations and enlargements. ![]() An image will reflect through a line, known as the line of reflection. A reflection close reflection A reflection is a mirror image of the shape.
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