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In this lesson, we will learn

- what is dilation or enlargement and reduction?
- dilation with scale factor greater than 1.
- dilation with scale factor between 0 and 1.
- dilation with a negative scale factor.
- dilation on the coordinate plane.

More Geometry Lessons

A** dilation** is a
transformation that produces an image that is the same shape as
the original, but is a different size. (The image is similar to
the original object). 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 center
of dilation.

The distance the points move depends on the scale factor.

In dilation, the center 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 center at O and scale factor

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

* Example* :

Enlarge triangle PQR with O as the center 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 center 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 center 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.

This video explores what happens when the scale factor or the center of the dilation is changed.

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 center 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.

Dilation of a Geometric Figure

This video will show you how to create a dilation of a geometric
figure using a center of dilation and a scale factor of half.

If the scale factor of a dilation is negative then the image will
be on the opposite side of the center of dilation compared with
the object.

This video will show you how to create a dilation of a geometric
figure using a center of dilation and a negative scale factor. (−
2)

We will now look at how to create a dilation on a coordinate plane.

DilationsA dilation is a non-rigid transformation, which means that the original and the image are not congruent. They are, however, similar figures. To perform dilations, a scale factor and a center of dilation are needed. If the scale factor is larger than 1, the image is larger than the original; if the scale factor is less than 1, the image is smaller than the original.

These videos show how to get the scale factor given the points of the shape and its image.

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