Triangle ABC is transformed to produce triangle DEF
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 rotation
To guarantee triangles ABC and DEF are congruent, the following transformations need to be applied in sequence:
a reflection, then a rotation
To determine which transformations guarantee congruence between triangles ABC and DEF, we need to understand the properties of congruent triangles.
Congruent triangles have the same shape and size. This means that corresponding sides are equal in length, and corresponding angles are equal in measure.
Now let's analyze each option:
a) A dilation, then a translation: This transformation changes the size of the triangle and then moves it horizontally or vertically. Since the sides and angles of the original triangle ABC would change, this transformation does not guarantee congruence.
b) A reflection, then a dilation: This transformation first reflects the triangle and then changes its size. The reflection by itself does not guarantee congruence because it reverses the orientation of the triangle. Therefore, this transformation does not guarantee congruence.
c) A reflection, then a rotation: This transformation first reflects the triangle and then rotates it. A reflection changes the orientation of the triangle, whereas a rotation does not. Hence, this transformation can potentially produce congruent triangles.
d) A rotation, then a translation: This transformation first rotates the triangle and then moves it horizontally or vertically. A rotation preserves the shape and size of the triangle, while the subsequent translation does not alter these properties. Therefore, this transformation guarantees congruence.
e) A translation, then a reflection: This transformation changes the position of the triangle and then reflects it. A translation does not change the shape or size, but the subsequent reflection reverses the orientation. Hence, this transformation does not guarantee congruence.
Based on the analysis, the transformation that guarantees congruence between triangles ABC and DEF is:
d) A rotation, then a translation.