Solved examples: Proving similarity using transformations (Grade 8 math)

📚 Quick Study Guide

• 📐 Similarity Transformations: Similarity transformations are transformations that preserve shape but not necessarily size. They include translations, rotations, reflections, and dilations.
• 🌐 Dilation: A dilation is a transformation that enlarges or reduces a figure by a scale factor. If the scale factor is greater than 1, the figure is enlarged. If the scale factor is between 0 and 1, the figure is reduced.
• 🚶 Translation: A translation is a transformation that slides a figure without changing its orientation or size.
• 🔄 Rotation: A rotation is a transformation that turns a figure around a fixed point.
• зеркало Reflection: A reflection is a transformation that flips a figure over a line.
• 🔗 Proving Similarity: To prove two figures are similar using transformations, show that a sequence of translations, rotations, reflections, and dilations maps one figure onto the other.
• 📏 Scale Factor: The ratio of corresponding side lengths in similar figures is called the scale factor.

Practice Quiz

1. Question 1: Which transformation does NOT preserve size?
   1. Translation
   2. Rotation
   3. Dilation
   4. Reflection
2. Question 2: If triangle ABC is dilated by a scale factor of 2 to create triangle A'B'C', what is the relationship between the side lengths of the two triangles?
   1. A'B' = AB
   2. A'B' = 2 * AB
   3. A'B' = 0.5 * AB
   4. A'B' = AB + 2
3. Question 3: Which sequence of transformations proves that triangle PQR is similar to triangle P'Q'R'?
   1. Translation only
   2. Rotation followed by reflection
   3. Dilation only
   4. Dilation followed by translation
4. Question 4: A figure is reflected over the x-axis. Does this transformation prove similarity?
   1. Yes
   2. No
   3. Only if followed by a dilation
   4. Only if the original figure is a square
5. Question 5: Triangle XYZ is rotated 90 degrees clockwise. How does this affect its similarity to the original triangle?
   1. It is no longer similar.
   2. It remains similar.
   3. It becomes congruent.
   4. The side lengths change.
6. Question 6: A quadrilateral is translated 5 units to the right and 3 units up. Does this transformation prove similarity?
   1. Yes
   2. No
   3. Only if the quadrilateral is a rectangle
   4. Only if followed by a dilation
7. Question 7: If two triangles have corresponding angles that are equal, and corresponding sides are in proportion, are the triangles similar?
   1. Yes
   2. No
   3. Only if they are right triangles
   4. Only if the scale factor is 1