Select ALL of the transformations that would guarantee triangles ABC and DEF are congruent.
a dilation, then a translation
a reflection, then a dilation
a reflection, then a rotation
a rotation, then a translation
a translation, then a reflection
a reflection, then a translation
a reflection, then a rotation
a rotation, then a translation
a translation, then a reflection
The transformations that would guarantee triangles ABC and DEF are congruent are:
a reflection, then a rotation
a rotation, then a translation
To determine which transformations would guarantee that triangles ABC and DEF are congruent, we need to review the definition of congruent triangles and the properties of different transformations.
Congruent triangles have the same size and shape. In other words, all corresponding angles are equal, and all corresponding sides are equal in length.
1. A dilation is a transformation that changes the size of a figure without changing its shape. A dilation can make two triangles similar, but not necessarily congruent. Therefore, a dilation followed by a translation would not guarantee congruence.
2. A reflection is a transformation that "flips" a figure over a line of reflection. If we apply a reflection followed by a dilation, we can change the size and shape of the triangles, which would not guarantee congruence.
3. Similarly, a reflection followed by a rotation can change the size and shape of the triangles, so it would not guarantee congruence either.
4. A rotation is a transformation that turns a figure around a fixed point. If we apply a rotation followed by a translation, we can change the position of the triangles, but not their size or shape. Therefore, this sequence of transformations would guarantee congruence.
5. Lastly, a translation is a transformation that slides a figure without changing its shape and size. If we apply a translation followed by a reflection, we can change the position and the orientation of the triangles, but not their size or shape. Therefore, this sequence of transformations would not guarantee congruence.
From the given options, the transformation sequence that would guarantee triangles ABC and DEF to be congruent is:
- a rotation, then a translation.