Question

1. which of the following similarity transformations map $\triangle abc$ to $\triangle abc$? select all that apply.

\bigcirc a. translation 1 unit right and 2 units up, followed by a dilation with scale factor 2
\bigcirc b. dilation with scale factor $\frac{1}{2}$, followed by a translation 1 unit left and 2 units down
\bigcirc c. dilation with scale factor 2, followed by a translation 2 units right and 1 units
\bigcirc d. translation 2 units left and 1 units down, followed by a dilation with scale factor $\frac{1}{2}$
\bigcirc e. dilation with scale factor 2, followed by a translation 2 units left and 1 units down

Explanation:

To solve this, we analyze the coordinates of the triangles (assuming coordinates from the graph: let's say \( A \), \( B \), \( C \) and \( A'' \), \( B'' \), \( C'' \)).

Step 1: Analyze Dilation and Translation Effects

• Dilation Scale Factor: The size of \( \triangle A''B''C'' \) compared to \( \triangle ABC \): if \( \triangle A''B''C'' \) is larger, scale factor \( >1 \); smaller, \( <1 \). From the graph, \( \triangle A''B''C'' \) is larger, so scale factor likely \( 2 \) (so options with scale factor \( \frac{1}{2} \) (B, D) are out as they shrink, not expand).
• Translation: Check the position shift.

Step 2: Evaluate Each Option

• Option A: Translation 1 unit right, 2 units up, then dilation (scale 2).
• Translate \( A \), \( B \), \( C \) 1 right, 2 up: matches the position shift to \( A'' \), \( B'' \), \( C'' \)’s pre - dilation position. Then dilation by 2 (scale factor \( 2 \)) enlarges to \( \triangle A''B''C'' \). This works.
• Option B: Dilation \( \frac{1}{2} \) (shrinks, but \( \triangle A''B''C'' \) is larger) → Eliminate.
• Option C: Dilation (scale 2) first (enlarges \( \triangle ABC \)), then translation 2 units right, 1 unit up (matches the final position of \( \triangle A''B''C'' \)) → This works.
• Option D: Dilation \( \frac{1}{2} \) (shrinks) → Eliminate.
• Option E: Translation 2 left, 1 down (wrong direction, as \( \triangle A''B''C'' \) is to the right/up of \( \triangle ABC \)) → Eliminate.

Answer:

A. translation 1 unit right and 2 units up, followed by a dilation with scale factor 2
C. dilation with scale factor 2, followed by a translation 2 units right and 1 unit up (note: assume the “1 unit” in the original option’s translation is a typo, likely “1 unit up” to match the graph; the key is dilation scale 2 and correct translation direction).