Question

Translate Triangle A by vector (3, -1) to give triangle B.

Then, rotate your triangle B 180° around the origin to give triangle C.
Describe fully the SINGLE transformation that maps triangle A onto triangle C.

Answer 1

The single transformation that maps Triangle A onto Triangle C is a 180° rotation around a point that is offset from the origin by the translation vector (3, -1).

Step-by-step explanation:

When Triangle A is first translated by the vector (3, -1), and subsequently rotated 180° around the origin, the resulting transformation that maps Triangle A onto Triangle C can be described in simpler terms. This combined transformation is, in fact, a rotation of 180° around a point that is the translation vector (3, -1) away from the origin.

The reasoning behind this is that translating first and then rotating 180° is equivalent to a single 180° rotation about a point not at the origin. To determine the location of this center of rotation, imagine the translation moving the origin to the new center, from which the 180° rotation then takes place.

Answer 2

The single transformation that maps triangle A onto triangle C is a **rotation of 180 degrees around a point that is offset from the origin by the translation vector (-3, 1)**.

Here's how we can arrive at this answer:

1. **Translation:** Triangle A is first translated by the vector (-3, 1), which moves each point of the triangle 3 units to the left and 1 unit up. This results in triangle B.

2. **Rotation:** Triangle B is then rotated 180 degrees around a fixed point. Since the final triangle, C, is the mirror image of A across the vertical axis, this fixed point must be offset from the origin by the translation vector.

3. **Combined Transformation:** Therefore, the single transformation that achieves the same result as the two separate steps is a 180-degree rotation around a point that is 3 units to the left and 1 unit up from the origin.

In other words, imagine fixing a point at (-3, 1) and then placing a sheet of paper with triangle A on top of it. Rotating the paper 180 degrees around this point will make triangle A coincide with triangle C.