Transformations are important when we study shapes in geometry. They help us understand things like similarity and congruence. Here are the main types of transformations that keep the shape and size the same:

1. Translation:

   • A translation moves a shape from one place to another without changing its shape or size.
   • The distance between matching points stays the same, which keeps the figure unchanged compared to where it started.

2. Rotation:

   • A rotation turns a shape around a fixed point, called the center of rotation.
   • The angles and lengths of the sides do not change. For example, if you rotate a triangle by 90 degrees around one of its corners, you still have a triangle that is exactly the same.

3. Reflection:

   • A reflection flips a shape over a line, which is called the line of reflection. This creates a mirror image of the original shape.
   • The distance between matching points is still the same, so the two figures are congruent. For example, if you reflect a rectangle over one of its sides, you'll still have a rectangle that is exactly the same.

Keeping Shape and Size the Same

• Congruence:

  • All three transformations (translation, rotation, reflection) give us figures that are congruent. This means that all the corresponding sides and angles are equal.

• Mathematical Representation:

  • If we take a shape ( A ) and change it to shape ( B ), we can say ( A \cong B ) to show they are congruent.
  • For translations, if point ( A(x, y) ) moves to point ( B(x+a, y+b) ), the shape and size stay the same. This means ( |AB| = |A'B'| ).

• Facts About Transformations:

  • In real-life geometry, transformations are commonly used in problems about congruence. For example, using transformations in geometry proofs can help us make valid conclusions in over 75% of congruence examples.

These transformations help us keep the properties of shapes intact. This gives us a better understanding of how shapes relate to each other in the study of similarity and congruence.