IXL - Similar triangles and similarity transformations (Geometry practice)

You can transform

△

by translating it and then performing a dilation centered at the origin. So,

△

Find the translation rule and the scale factor of the dilation.

Simplify the scale factor and write it as a proper fraction, improper fraction, or whole number.

Translation:
(
x
,
y
)
↦
i
,
i
Scale factor:

ref_doc_title.

Learn with an example

question

You can transform

△

by translating it and then performing a dilation centered at the origin. So,

△

Find the translation rule and the scale factor of the dilation.

Simplify the scale factor and write it as a proper fraction, improper fraction, or whole number.

Translation:
(
x
,
y
)
↦
i
,
i
Scale factor:

key idea

The origin is the point (0, 0).

The image of the point

(

)

translated

units horizontally and

units vertically is

(

)

is positive the point is translated to the right and if

is negative the point is translated to the left. If

is positive the point is translated up and if

is negative the point is translated down.

The image of the point

(

)

dilated with a scale factor of

centered at the origin is

(

)

solution

△

is a translation and dilation of

△

This means that

corresponds to

and

Since the dilation is centered at the origin, make sure that one pair of corresponding vertices is located at the origin before you apply the dilation. This makes sure that the dilation will not affect the coordinates of these points. (In general, it is always the case that the center of the dilation is not affected by the dilation.)

From the diagram, you can see that

is already located at the origin. So, find the translation that maps

(

)

(

)

To translate

(

4, 9) to

(0,

0), move 4 units to the left and 9 units down. So, the translation that maps

is given by the rule

(

)

↦

(

–

)

Next, apply this translation rule to the three vertices of

△

The image of

the image of

lies on top of

and the image of

lies on top of

Now, find the scale factor of the dilation that will expand the image of

△

onto

△

Choose a pair of corresponding vertices not located at the origin, like

(

–

)

and the image of

after the translation, which is

(

–

)

Calculate the ratio of the

-coordinates

or the ratio of the

-coordinates.

For instance, the ratio of the

-coordinates

so the scale factor of the dilation that maps

(

–

)

onto

(

–

)

is 5. This dilation also maps the image of

which is

(

–

)

onto

(

–

)

and fixes both

and

In summary, the translation and scale factor are:

Translation:
(
x
,
y
)
↦
(
x
–
4
,
y
–
9
)
Scale factor: 5

Looking for more? Try these:

Lesson: Translations

Lesson: Dilations

Lesson: Similarity

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Teacher tools

Work it out

Not feeling ready yet? These can help:

H.4Translations: write the rule

I.4Dilations: find the scale factor

M.1Ratios in similar figures

Lesson: Translations

Lesson: Dilations

Lesson: Similarity

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