Question

ΔABC was dilated from the origin by a scale factor of 2 to create ΔA'B'C'. Which statement explains why the triangles are similar? a) Dilations preserve side length; therefore, the corresponding sides of ΔABC and ΔA'B'C' are congruent. b) Dilations preserve betweenness of points; therefore, the corresponding sides of ΔABC and ΔA'B'C' are congruent. c) Dilations preserve orientation; therefore, the corresponding angles of ΔABC and ΔA'B'C' are congruent. d) Dilations preserve angle measure; therefore, the corresponding angles of ΔABC and ΔA'B'C' are congruent.

Answer 1

Triangles ΔABC and ΔA'B'C' are similar because dilations preserve angle measures, which is a requirement for triangle similarity as per the Angle-Angle (AA) similarity postulate.

Step-by-step explanation:

The question asks to determine why triangles ΔABC and ΔA'B'C', which were created through a dilation from the origin with a scale factor of 2, are similar. The correct statement that explains why the triangles are similar is (d) Dilations preserve angle measure; therefore, the corresponding angles of ΔABC and ΔA'B'C' are congruent.

Dilations are transformations that produce a figure of the same shape but a different size, which results in similar triangles. Since all corresponding angles remain equal and the sides are proportional, we can confirm the similarity of the triangles as per the Angle-Angle (AA) similarity postulate.

Options (a) and (b) are incorrect because dilations do not preserve side length or betweenness of points but rather change distances in a manner proportional to the scale factor. Option (c) could be mistaken for being correct since dilations do preserve orientation, but it is not the defining reason for the similarity of the shapes. The preservation of angle measures is central to establishing similarity in dilated figures.

Answer 2

Dilations preserve orientation; therefore, the corresponding angles of ΔABC and ΔA'B'C' are congruent.

c) Dilations preserve orientation; therefore, the corresponding angles of ΔABC and ΔA'B'C' are congruent.

Dilations are transformations that resize figures while keeping the same shape. When a figure, in this case, triangle ΔABC, is dilated from the origin by a scale factor of 2 to create triangle ΔA'B'C', the corresponding angles of the two triangles will be congruent. This is because dilations preserve the orientation of the figure, meaning that the angles in the original triangle will have the same measure as the corresponding angles in the dilated triangle.

Option (a) mentions that dilations preserve side length, but this is not necessarily true for similarity. Option (b) talks about betweenness of points, which is not a characteristic preserved by dilations. Option (d) mentions angle measure, which is correct, but it is more accurate to say that dilations preserve orientation, leading to congruent corresponding angles.