What is the inverse of a translation, or composition of translations?[br]1. Click on the checkbox to begin, and create a translation for triangle ABC by dragging the brown vector. Note that a or b or both can be negative.[br]2. Click on the second checkbox and drag the red vector to create the translation that maps triangle A'B'C' back to its pre-image, ABC. What are the values for c and d?[br]3. Reset the vectors to their original positions, (0, 1), and click on the third checkbox.[br]4. Drag the brown vector to create translation T. Drag the red vector for T-squared. Note that T-squared means the translation T applied twice, first to the pre-image ABC, and then to the image of ABC under T. What are the values for c and d?[br]5. Drag the orange vector to map the second image back to the pre-image ABC. What are the values of f and g?

The inverse of a translation adds the opposite values to each of the coordinates so that the composition of the translation, T, composed with T-inverse, results in the identity. Verify that this is commutative.[br][br]As you know, the composition of a translation is the sum of the translated coordinates, so that the inverse of the composition is the sum of the opposites of the coordinates, resulting in the identity.