Question

On a coordinate plane, 2 rectangles are shown. The first rectangle has points A (negative 2, negative 1), B (4, negative 1), C (4, negative 3), D (negative 2, negative 3). The second rectangle has points A prime (negative 5, 1), B prime (1, 1), C prime (1, negative 1), D prime (negative 5, negative 1).
Irina wants to tile her floor using the translation shown below. Which is the rule for this translation?

T–3, 2(x, y)
T2, –3(x, y)
T3, 2(x, y)
T–2, 3(x, y)

Answer 1

To determine the correct translation rule for the transformation from the first rectangle to the second rectangle, we can analyze the coordinates of the corresponding vertices.

1. **First Rectangle**:
- A (-2, -1)
- B (4, -1)
- C (4, -3)
- D (-2, -3)

2. **Second Rectangle**:
- A' (-5, 1)
- B' (1, 1)
- C' (1, -1)
- D' (-5, -1)

Now, let's find the translation vector that takes a point \((x, y)\) in the first rectangle to the corresponding point in the second rectangle.

For point A to A':
- From A (-2, -1) to A' (-5, 1)
- The change in the x-coordinate: \(-5 - (-2) = -5 + 2 = -3\)
- The change in the y-coordinate: \(1 - (-1) = 1 + 1 = 2\)

This indicates a translation of \((-3, 2)\).

Now let's apply the translation rule \(T_{-3, 2}(x, y)\):
\[
(x, y) \to (x - 3, y + 2)
\]

Let's confirm this translation with another point. For point B to B':
- B (4, -1) to B' (1, 1)
- Change in x-coordinate: \(1 - 4 = -3\)
- Change in y-coordinate: \(1 + 1 = 2\)

This confirms the translation is consistent for another point.

Thus, the translation rule that transforms the first rectangle to the second rectangle is:

**T–3, 2(x, y)**