Isometry is a transformation that preserves distances.[br][br][b]Definition:[/b] A transformation [math]T[/math] of the plane is an [b]isometry[/b] if, for any two points [math]P[/math] and [math]Q[/math], the distance between their images [math]P'[/math] and [math]Q'[/math] is equal to the distance between [math]P[/math] and [math]Q[/math].[br]The shape of the image must remain congruent to the original object.[br][br]Every isometry in a 2D plane can be categorized into one of four types.[br][br][b]A. Translation[/b][br]A translation moves every point in the plane a same distance in a same direction. It is defined by a vector.[br]Use the grid box below to create a point and perform a translation using a vector of your choice.[br]Next, repeat it by creating a shape.[br]Observe its property.

[b]B. Rotation[/b][br]A rotation turns the plane around a fixed point (the center) by a specific angle.[br]Use the grid box below to create a point and perform a rotation by an angle of your choice.[br]Next, repeat it with a shape.[br]Observe its property.[br][br][b]C. Reflection[/b][br]A reflection "flips" the plane across a fixed line (the axis of reflection).[br]Use the grid box below to create a point and perform a reflection on a line of your choice.[br]Next, repeat it with a shape.[br]Observe its property.[br]Hint: If you label the vertices of a triangle A, B, C clockwise, their images A', B', C' will appear counter-clockwise.[br][br][b][size=150]Tip![/size][/b][br][size=85]How to create an isometry transformation on the grid.[br][list=1][*]Create a point: Click on 'point' tool and mark a point on the grid.[/*][*]Translation: Click on 'translate by vector' tool, select the point made earlier, then mark another two points to indicate the vector.[/*][*]Rotation: Click on 'rotate about point' tool, select the point made earlier, then mark a point for rotation, input an angle value, and click 'enter'.[/*][*]Reflection: Click on 'line' tool', mark two points to draw a line. Click on 'reflect on line' tool, select the point made earlier, then click on the drawn line.[/*][*]Create a shape: Click on 'polygon' tool, mark the vertices to form a shape.[/*][/list][/size]

[b]Self-Directed Task[/b] (Use the grid below)[br][list=1][*]Create a segment [math]AB[/math]. Apply a rotation and then a reflection. Measure the new segment [math]A''B''[/math]. Does it match the original length?[/*][*]Create any shape. Can you create a single reflection that results in the same image as two translations?[/*][*]Create a triangle. Reflect it over a horizontal line. Then, translate it +3 units along the [math]x-[/math]axis. Try to achieve this same result using only one of the other three isometries.[/*][/list]

You will realise it is not possible.[br]This is why Glide Reflection is a distinct fourth type of isometric transformation.[br][b][br]D. Glide Reflection[/b][br]A glide reflection is a composition of a reflection and a translation, where the translation vector is parallel to the line of reflection.[br][br]Analogy: Think of footprints in the sand. To get from a left footprint to a right footprint, you must reflect it across a center line and then slide it forward.[br][br]Try this out:[br][list=1][*]Create a shape (object) on a grid.[/*][*]Translate / Slide your shape according to a given vector. Lightly sketch the image prime (the first transformation image).[/*][*]Reflect / Flip the image prime over the reflection line. Ensure the reflection line is parallel to the vector in Step 2.[/*][*]Now, repeat the transformation for the same shape at its original position but perform the reflection followed by translation.[/*][*]Observe its property.[/*][/list]

[list=1][*]From the exploration above, can you summarise which type of isometric transformation preserves its orientation and which does not?[/*][*]If a transformation stretches a square into a rectangle, is it an isometry? Why or why not?[/*][/list][br][br][br][i][size=100][size=85]Answer: [br][list=1][*][i][size=100][size=85]Translation and rotation preserve orientation; reflection and glide reflection reverse the orientation.[br][/size][/size][/i][/*][*][i][size=100][size=85]No, it is not an isometry. By definition, an isometry must preserve distance between all points. If a square is stretched into a rectangle, the side lengths (distances) change, and the object is no longer congruent to the original shape. This is known as a dilation or enlargement.[/size][/size][/i][/*][/list][/size][/size][/i]