Solving problems using affine transformations

Affine transformations are functions that preserve parallelism and ratios of lengths along parallel lines. They include translations, rotations, scalings, shears, and their combinations. In problem solving, we can use affine transformations to simplify geometric figures, making it easier to analyze or solve problems.

Important properties

• Affine transformations map lines to lines and preserve parallelism.

• Midpoints and ratios along parallel lines are preserved.

• Shapes like parallelograms remain parallelograms, but angles and lengths may change (except for ratios on parallel lines).

• We can use affine transformations to 'normalize' a figure (for example, mapping a triangle to an easier shape like an isosceles or right triangle) to simplify calculations.