Question

lennox was curious if triangles $\triangle abe$ and $\triangle dce$ were similar, so she tried to map one figure onto the other using a reflection and a dilation.

lennox concluded:
\its not possible to map $\triangle dce$ onto $\triangle abe$ using a sequence of rigid transformations and dilations, so the triangles are not similar.\
what error did lennox make in her conclusion?
choose 1 answer:
\\(\boldsymbol{\text{a}}\\) one more transformation — a horizontal stretch — would map $\triangle dce$ onto $\triangle abe$. so the triangles are similar.
\\(\boldsymbol{\text{b}}\\) one more transformation — a dilation — would map $\triangle dce$ onto $\triangle abe$. so the triangles are similar.
\\(\boldsymbol{\text{c}}\\) there is no error. this is a correct conclusion.

Explanation:

To determine the error in Lennox's conclusion, we recall the definition of similar triangles: two triangles are similar if one can be mapped onto the other using a sequence of rigid transformations (reflection, rotation, translation) and a dilation.

• Analyzing Option A: A horizontal stretch is not a dilation (dilation is a uniform scaling, while a stretch is a non - uniform transformation and not part of the similarity transformation rules). So this option is incorrect.
• Analyzing Option B: If we perform a dilation on $\triangle DCE$, we can adjust its size to match $\triangle ABE$ after the reflection (a rigid transformation). Since we can use a rigid transformation (reflection) and a dilation to map $\triangle DCE$ onto $\triangle ABE$, the triangles are similar. Lennox made an error by not considering that a dilation (along with the reflection) can map one triangle to the other.
• Analyzing Option C: Since we can map the triangles using a reflection and a dilation, Lennox's conclusion that they are not similar is wrong, so this option is incorrect.

Answer:

B. One more transformation — a dilation — would map $\triangle DCE$ onto $\triangle ABE$. So the triangles are similar.