Isometry
A transformation is the movement of an object within a plane. The original figure is called the pre-image, and the transformed figure is called the image. There are four major types of transformations:
1) Translation 2) Reflection
A transformation is the movement of an object within a plane. The original figure is called the pre-image, and the transformed figure is called the image. There are four major types of transformations:
1) Translation 2) Reflection
3) Rotation 4) Dilation
The first three on the list above are isometries. An isometry is a congruence transformation. It preserves the shape and size of the figure by preserving segment lengths and angle measures. A dilation is not an isometry, and therefore will be explored more thoroughly in the similarity unit. There is another isometry that we will explore later in this unit called a glide reflection. If the point \(A\) is our pre-image, then we usually use a notation of \(A'\) as the image.
Translation:
A translation moves every point of a pre-image the same distance and direction to create the image. There is no reflection, rotation, or change in size. Since there is no change in size or shape, translations are isometries. A translation can be defined using coordinate notation or using a vector.
Translation:
A translation moves every point of a pre-image the same distance and direction to create the image. There is no reflection, rotation, or change in size. Since there is no change in size or shape, translations are isometries. A translation can be defined using coordinate notation or using a vector.
A vector is similar to a ray, except that it has a beginning and ending point, so it does not go on forever. The point where the vector begins, in this case, point \(C\), is called the initial point. The point where the vector ends, in this case point \(C\), is called the terminal point. A vector has both magnitude (length) and direction (down \(3\) and right \(7\)). A vector is named by the initial point, then the terminal point, and cannot be named backwards. Vectors can be written in component form or using coordinate notation. This translation can be described by \(\overrightarrow{CC'}\), which has the component form of \(\big\langle 7, -3\big\rangle\). Or this translation can be described using coordinate notation: \((x, y)\rightarrow (x + 7, y - 3)\).
Use the app below to see how the vector translates the triangle.
Use the app below to see how the vector translates the triangle.
Example 2:
Find the image, \(Z’\), of point \(Z(2, 4)\) when it is translated using the rule \((x, y)\rightarrow (x - 3, y + 5)\).
Solution:
\(Z’\ = (2 - 3, 4 + 5) = (-1, 9)\)
Example 3:
Find the pre-image, \(B\), of point \(B’ (4, 3)\) when the point has been translated by the \(\overrightarrow{AC} = \big\langle 6, -1\big\rangle\).
Find the image, \(Z’\), of point \(Z(2, 4)\) when it is translated using the rule \((x, y)\rightarrow (x - 3, y + 5)\).
Solution:
\(Z’\ = (2 - 3, 4 + 5) = (-1, 9)\)
Example 3:
Find the pre-image, \(B\), of point \(B’ (4, 3)\) when the point has been translated by the \(\overrightarrow{AC} = \big\langle 6, -1\big\rangle\).
Quick Check
1) What is the vector that translates the point \(P(-13, -5)\) to \(P'(-1, 9)\)?
2) What are the coordinates of \(\triangle SUB\) after the translation of \((x, y)\rightarrow (x - 6, y - 1)\)?
1) What is the vector that translates the point \(P(-13, -5)\) to \(P'(-1, 9)\)?
2) What are the coordinates of \(\triangle SUB\) after the translation of \((x, y)\rightarrow (x - 6, y - 1)\)?