We are given two triangles.

We will determine if the given two figures are similar by using transformations. Let's first recall the definition of similar figures.

Similar Figures
Two figures are similar if they have the same shape and the ratios of their corresponding linear measures are equal.

We need to determine whether the figures have the same shape by finding a sequence of transformations that maps one figure onto the other. Notice that the orientations of the triangles are different, so one of the transformations will be a reflection.

Reflection
A transformation in which every point of a figure is reflected in the line of reflection

Let's first find the line of reflection that reflects $△ D E F$ onto $△ A B C .$ We will make sure that the directions of the image of the reflection and $△ A B C$ will be the same and that the image of vertex $F$ overlaps with vertex $C$ of $△ A B C .$ Remember that the line of reflection is equidistant to both preimage and image points.

The distance between points $F$ and $C$ is $2$ units, so the line of reflection will be $1$ unit away from both points. When we reflect $△ D E F$ over the vertical line, the orientation of the reflected image will be the same as $△ A B C .$ This suggests that the line of reflection will be a vertical line that is $1$ unit away from both $F$ and $C .$ Let's proceed with the reflection!

Next, since the size of the triangles are not the same, we will determine whether we can perform a dilation that maps the image of the reflection onto

△ A B C .

To help with this, we need to find the ratios of the corresponding sides.

\frac{F ^{'} E ^{'}}{C B} = \frac{3}{6}, \frac{E ^{'} D ^{'}}{B A} = \frac{3}{6} \frac{D ^{'} F ^{'}}{A C} = \frac{3}{6}

Since the ratios of the corresponding sides of

△ D E F

and

△ A B C

are equal, a dilation can map the image of the reflection onto

△ A B C .

From here, we can conclude that a reflection followed by a dilation can map

△ D E F

onto

△ A B C .

Therefore, these figures are similar.

Exercise