Question

question 1-4
if $\triangle abc \sim \triangle pqr$ below, what combination of transformations maps $\triangle abc$ onto $\triangle pqr$? select all that apply
image of coordinate grid with triangles abc and pqr
a. yes, translation 4 units to the right, rotation of $270^\circ$ counterclockwise about $c$, and a dilation centered at $c$ with a scale factor of 2
b. yes, translation 3 units down, rotation of $180^\circ$ counterclockwise about $b$, and a dilation centered at $b$, and a scale factor of 2
c. yes, translation 4 units down, reflection in the y - axis, and a dilation centered at $b$ with scale factor of 2
d. yes, dilation centered at the origin with a scale factor of 2 followed by a rotation of $180^\circ$ counterclockwise about the origin.
e. no, the figures are not similar because the sequence of transformations does not preserve angle measure.

Explanation:

To solve this, we analyze each option:

Option A:

• Translate 1 unit right, rotate \(270^\circ\) counterclockwise about \(C'\), then dilate (scale factor 2) about \(C'\).
• Translations, rotations, and dilations preserve similarity (angles, proportional sides). This sequence can map \(\triangle ABC\) to \(\triangle PQR\) as \(\triangle ABC \cong \triangle PQR\) (so similar with scale factor 2).

Option B:

• Rotation of \(180^\circ\) counterclockwise about \(B'\) would flip the triangle, but the orientation of \(\triangle PQR\) relative to \(\triangle ABC\) does not match a \(180^\circ\) rotation here. So this is incorrect.

Option C:

• Reflection over the \(y\)-axis would flip the triangle horizontally, but \(\triangle PQR\)’s orientation does not match a reflection. Incorrect.

Option D:

• Dilation (scale 2) then \(180^\circ\) rotation: A \(180^\circ\) rotation after dilation would flip the triangle, but the final position of \(\triangle PQR\) does not match this. Incorrect.

Option E:

• Similarity transformations (translation, rotation, dilation) preserve angle measures and side proportions. Since \(\triangle ABC \cong \triangle PQR\), they are similar. So this is incorrect.

Answer:

A. Yes, translation 1 unit to the right, rotation of \(270^\circ\) counterclockwise about \(C'\), and a dilation centered at \(C'\) with a scale factor of 2