Question

Which of the following represents the translation of A (1, −2) along the vector <−5, 1> and then the vector <3, 0>?
1.) A (1, −2) → A ′(2, −7) → A ″(6, −7)
2.) A (1, −2) → A ′(−5, −2) → A ″(−15, 0)
3.) A (1, −2) → A ′(−5, 1) → A ″(3, 0)
4.) A (1, −2) → A ′(−4, −1) → A ″(−1, −1)

1. The points that represent the translations of the point A(x, y) = (1, – 2) are A'(x, y) = (- 4, – 1) and A”(x, y) = (- 1, – 1). (Correct choice: 4)

### What is the resulting point by applying translations?

Rigid transforations are transformations applied on geometric loci such that Herein we know the coordinates of a point on a Cartesian plane, on which two translation vectors, a kind of rigid transformation, are applied to determine the coordinates of the resulting point according to the following formula:
P'(x, y) = P(x, y) + T₁(x, y)       (1)
P”(x, y) = P'(x, y) + T₂(x, y)     (2)
Where:
• P(x, y) – Original point
• P'(x, y), P”(x, y) – Resulting points.
• T₁(x, y), T₂(x, y) – Translation vectors.
If we know that A(x, y) = (1, – 2), T₁(x, y) = (- 5, 1) and T₂(x, y) = (3, 0), then the resulting points are:
A'(x, y) = (1, – 2) + (- 5, 1)
A'(x, y) = (- 4, – 1)
A”(x, y) = (- 4, – 1) + (3, 0)
A”(x, y) = (- 1, – 1)
Then, the points that represent the translations of the point A(x, y) = (1, – 2) are A'(x, y) = (- 4, – 1) and A”(x, y) = (- 1, – 1). (Correct choice: 4)