In which case does the transformation of \( \triangle Q R S \) result in an image \( \triangle D E F \) where \( \angle Q \cong \angle D, \angle R \cong \angle E, \angle S \cong \angle F \), and \( \frac{Q R}{D E}=\frac{Q S}{D F}=\frac{R S}{E F} \) a translation of 6 units to the left and 8.5 units up followed by a reflection over the line \( y=2 x \) a reflection over the line \( y=-2 x \) 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 \( A \)

To determine which transformation results in the image \( \triangle DEF \) such that: - \( \angle Q \cong \angle D \) - \( \angle R \cong \angle E \) - \( \angle S \cong \angle F \) - \( \frac{QR}{DE} = \frac{QS}{DF} = \frac{RS}{EF} \) we need to identify a **similarity transformation**. Similarity transformations preserve the shape of the figure, meaning that all corresponding angles remain congruent and the sides are proportional by a consistent scale factor. Let's analyze each option: 1. **Translation followed by Reflection:** - **Transformation:** Translation (6 units left, 8.5 units up) followed by reflection over the line \( y = 2x \). - **Effect:** Both translation and reflection are **isometries** (distance and angle preserving transformations) with a scale factor of 1. - **Result:** The triangles remain congruent (\(\text{scale factor} = 1\)). 2. **Reflection followed by Translation:** - **Transformation:** Reflection over the line \( y = -2x \) followed by translation (6.5 units right, 3.5 units down). - **Effect:** Both reflection and translation are **isometries** with a scale factor of 1. - **Result:** The triangles remain congruent (\(\text{scale factor} = 1\)). 3. **Rotation followed by Dilation:** - **Transformation:** Rotation of \( 45^\circ \) clockwise about vertex \( A \) followed by dilation with a scale factor of 0.95 about the origin. - **Effect:** Rotation is an isometry, and dilation is a **similarity transformation** that changes the size by a scale factor of 0.95. - **Result:** The triangles are similar with corresponding sides in the ratio \( 0.95 \). 4. **Dilation followed by Rotation:** - **Transformation:** Dilation with a scale factor of 1 about the origin followed by rotation of \( 45^\circ \) clockwise about vertex \( A \). - **Effect:** Dilation with a scale factor of 1 is effectively an identity transformation (no change), and rotation is an isometry. - **Result:** The triangles remain congruent (\(\text{scale factor} = 1\)). **Conclusion:** Only option **3** introduces a change in scale factor while preserving the angle congruencies, making it the correct similarity transformation that satisfies the given conditions. **Answer:** A rotation of \( 45^{\circ} \) clockwise about vertex \( A \) followed by a dilation by a scale factor of 0.95 about the origin