Dilation Transformation

A dilation is a transformation that produces an image that is the same shape as the original, but is a different size. Dilation is a transformation in which each point of an object is moved along a straight line. The straight line is drawn from a fixed point called the centre of dilation.

The distance the points move depends on the scale factor.

In dilation, the centre of dilation is the only invariant point.

Scale factor =

If the scale factor is greater than 1, the image is an enlargement.

If the scale factor is between 0 and 1, the image is a reduction.

Example:

The figure shows two similar triangles PQR and P’Q’R’.

Triangle P’Q’R’ is a dilation of triangle PQR. We say that triangle PQR is transformed onto triangle P’Q’R’ by a dilation with centre at O and scale factor

Enlargement

We will first look at enlargements which are dilations with scale factors greater than 1

Example :

Enlarge triangle PQR with O as the centre of dilation and a scale factor of 2.

Solution:

Step 1: Measure OP.

Step 2: Extend the line OP to the point P’ such that OP’ = 2OP.

Step 3: Repeat the steps for all the vertices: point Q to get Q’ and point R to get R’.

Step 4: Join the points P’Q’R’ to form the image.

Example:

Enlarge triangle ABC with C as the centre of dilation and a scale factor of 3.

Solution:

Step 1: Measure CA.

Step 2: Extend the line CA to the point A’ such that CA’ = 3CA.

Step 3: Repeat the steps for point B to get B’.

Note that in this example, all the points in the triangle have been transformed except point C, which is the only invariant point.

Example:

Draw an image of the figure PQRS. O is the centre of dilation and the scale factor is 1.5.

Solution:

Step 1: Join OP.

Step 2: Extend the line OP to OP’, such that OP’ = 1.5 × OP

Step 3: Repeat for all the other vertices Q, R and S.

Step 4: Join P’, Q’, R’ and S’ to form the image.

Reduction

If the scale factor of a dilation is between 0 and 1, the image will be smaller than the object. It is then called a reduction.

Example :

Enlarge triangle PQR with O as the centre of enlargement and scale factor .

Solution:

Step 1 : Join O to P.

Step 2 : Mark off the point P ’ on OP such that OP ’ = OP.

Step 3 : Repeat the steps for all the vertices: point Q to get Q ’ and point R to get R ’ .

Step 4 : Join the points P’Q’R’ to form the image.

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