Let's analyze the transformations described in the image to determine which ones produce a congruent triangle when applied to triangle $\triangle ABC$.

Key Idea

A congruent triangle results when the transformation preserves the size and shape of the triangle. Such transformations include:

1. Reflections
2. Rotations
3. Translations

Transformations such as dilations (with a scale factor other than 1) do not preserve size and therefore do not produce congruent triangles.

Analysis of Options:

1. Reflection across $\overline{AB}$ followed by rotation 90° clockwise about point $A$:

   • A reflection preserves size and shape.
   • A 90° rotation also preserves size and shape.
   • Result: Congruent triangle.

2. Dilation centered at point $A$ with scale factor 0.5 followed by a translation 4 inches right:

   • Dilation with a scale factor of 0.5 reduces the size of the triangle, violating congruence.
   • Result: Not congruent.

3. Rotation 180° about point $B$ followed by translation 7 units to the left and 3 units down:

   • A 180° rotation preserves size and shape.
   • A translation also preserves size and shape.
   • Result: Congruent triangle.

4. Reflection across $\overline{BC}$ followed by dilation with scale factor 3 centered at point $C$:

   • Reflection preserves size and shape.
   • Dilation with a scale factor of 3 enlarges the triangle, violating congruence.
   • Result: Not congruent.

Final Answers:

The transformations that produce a congruent triangle are:

• Option 1: Reflection across $\overline{AB}$ followed by rotation 90° clockwise about $A$.
• Option 3: Rotation 180° about $B$ followed by translation 7 units to the left and 3 units down.

Would you like a more detailed explanation of any part? Let me know! 😊

Five Related Questions:

1. What is the mathematical definition of congruence in transformations?
2. How do reflections and rotations preserve congruence?
3. Can a dilation ever produce a congruent figure? Why or why not?
4. How do translations alone affect congruence?
5. What are real-life examples of congruence-preserving transformations?

Tip:

Always double-check whether transformations involve scaling (dilation), as they do not preserve congruence unless the scale factor is 1.