{{ stepNode.name }}

Proceed to next lesson

An error ocurred, try again later!

Chapter {{ article.chapter.number }}

{{ article.number }}. # {{ article.displayTitle }}

{{ article.introSlideInfo.summary }}

{{ 'ml-btn-show-less' | message }} {{ 'ml-btn-show-more' | message }} {{ 'ml-lesson-show-solutions' | message }}

{{ 'ml-lesson-show-hints' | message }}

| {{ 'ml-lesson-number-slides' | message : article.introSlideInfo.bblockCount}} |

| {{ 'ml-lesson-number-exercises' | message : article.introSlideInfo.exerciseCount}} |

| {{ 'ml-lesson-time-estimation' | message }} |

Image Credits *expand_more*

- {{ item.file.title }} {{ presentation }}

No file copyrights entries found

A dilation is a point transformation given by a center point $O$ and a scale factor $r$.
In this formula $O$ is the center of the dilation, $r$ is the scale factor, $A$ is the preimage, and $A_{′}$ is the image point. This is illustrated on the diagram. Move point $A$ and see how the image point moves.

$T:A↦A_{′} $

The image of each point is on the ray starting at the center and going through the preimage. The distance of the image from the center is calculated as follows. $OA_{′}=r⋅OA$

When the scale factor is greater than $1,$ the dilation is called an enlargement. When the scale factor is between $0$ and $1,$ the dilation is called a reduction.

The following is a list of a few essential properties of dilations.

- The dilation of a line is a line.

The formal proof of this property is beyond the scope of this course. The applet below, however, gives an experimental verification.

- Move the line, the center of dilation, and set the scale factor.
- Move a point along the line and verify that the image points indeed form a line parallel to the preimage.

- The dilation of a line segment is a line segment. The image is longer or shorter than the preimage in the ratio given by the scale factor.

The formal proof of this property is beyond the scope of this course. The applet below, however, gives an experimental verification.

- Move the segment, the center of dilation, and set the scale factor.
- Move a point along the segment and verify that the image points indeed form a segment parallel to the preimage.
- Check also that the ratio of the length of the image and the length of the preimage is indeed the scale factor. Note that the displayed lengths are only approximate, so the ratio will only be approximately equal to the scale factor.

- Dilations preserve angle measures.

Let $∠B_{′}A_{′}C_{′}$ be the dilated image of $∠BAC.$ According to the first property, $AC$ is parallel to $A_{′}C_{′}$ and $AB$ is parallel to $A_{′}B_{′}$

Let $M$ be the intersection point of $AB$ and $A_{′}C_{′},$ and focus on the parallel legs $AC$ and $A_{′}C_{′}.$ According to the Alternate Interior Angles Theorem, angles $∠MAC$ and $∠AMA_{′}$ are congruent.

Similarly, since the other legs are also parallel, angles $∠B_{′}A_{′}M_{′}$ and $∠AMA_{′}$ are congruent.

Since $∠AMA_{′}$ is congruent to both $∠BAC$ and $∠B_{′}A_{′}C_{′},$ the transitive property of congruence implies that these two angles are congruent.

By the definition of congruence, this completes the proof that dilation preserves angle measures.

$∠BAC≅∠B_{′}A_{′}C_{′}⇓m∠BAC=m∠B_{′}A_{′}C_{′} $

The methods to construct the image of a point depend on the scale factor.

1

Draw a ray from the center of dilation through the preimage point

Use a straightedge to draw a ray from the center of dilation through the preimage point.

2

Measure the distance from the center of dilation to the preimage point

Use a compass to measure the distance from the center of dilation to the preimage point.

3

Copy the measured distance on the ray

Keep the same opening of the compass and copy the distance on the ray. If the scale factor is $n,$ copy it $n−1$ times.

$OA_{′}=n×OA $

The diagram illustrates the construction for scale factor $3,$ so the distance is copied twice from the preimage point. To construct the image when the scale factor is $1/n,$ the properties of dilation can be used.

1

Draw two rays starting at the center of dilation

Use a straightedge to connect the center of dilation with the preimage point and draw a second ray starting at the center of dilation.

2

Copy the same distance on the second ray $n$ times

Use a compass to copy the same distance $n$ times on the second ray, starting at the center of dilation. The diagram illustrates the process when $n=3.$

3

Construct parallel lines

Mark the first and last point on the second ray. Connect the last point with the preimage point and construct a parallel line through the first point.

4

Mark the image point

The intersection point of this parallel line with the first ray is the constructed image.

$OA_{′}=n1 ×OA $

The combination of the previous two methods gives the dilation of a point by a scale factor $p/q.$

1

Dilation by a scale facor $1/q$

First, construct the image using a scale factor $1/q.$

$OT=q1 ×OA $

On the diagram $q=5.$

2

Dilation by a scale facor $p$

Copy the distance of this new point from the center of dilation $p$ times, starting at the center of dilation.

$OA_{′}=p×OTOT=q1 ×OA ⟹OA_{′}=qp ×OA $

The diagram below illustrates the dilation by scale factor $3/5$.

1

Construct the image of the vertices.

Depending on the scale factor, choose the appropriate method to construct the image of the vertices.

2

Connect the image vertices.

Segments are drawn between the image vertices to create the image polygon.