Question

in which case... and \\(\frac{qr}{de} = \frac{qs}{df} = \frac{rs}{ef} \
eq 1\\)?

• a translation of 6 units to the left and 8.5 units up followed by a reflection over the line \\(y = 2x\\)
• a reflection over the line \\(y = -2x\\) followed by a translation of 6.5 units to the right and 3.5 units down
• a rotation of \\(45^\circ\\) clockwise about vertex \\(a\\) followed by a dilation by a scale factor of 0.95 about the origin
• a dilation by a scale factor of 1 about the origin followed by a rotation of \\(45^\circ\\) clockwise about vertex \\(a\\)

Explanation:

Step1: Analyze given conditions

We need a transformation that preserves angle congruence ($\angle D \cong \angle R$, $\angle R \cong \angle E$, $\angle S \cong \angle F$) but results in side ratios $\frac{QR}{DE} = \frac{QS}{DF} = \frac{RS}{EF}
eq 1$. This means the transformation must be a similarity transformation (preserves angles) with a non-1 scale factor (changes size, so ratios ≠1).

Step2: Evaluate each option

• Option1: Translation + reflection: These are rigid motions (preserve size/shape, ratios=1). Eliminated.
• Option2: Reflection + translation: Rigid motions (preserve size/shape, ratios=1). Eliminated.
• Option3: Rotation + dilation: Rotation is rigid (preserves angles), dilation with scale factor 0.95 (≠1) changes size, so side ratios equal 0.95≠1, and angles stay congruent. Matches conditions.
• Option4: Dilation (scale=1) + rotation: Scale factor 1 means size is preserved, ratios=1. Eliminated.

Answer:

a rotation of 45° clockwise about vertex $A$ followed by a dilation by a scale factor of 0.95 about the origin