Question

A coordinate plane. Point A is plotted at (2, 7). Point B (2, 1). Point C (9, 8). Point D (8, 3). Point E (7, 7).
A point has been translated right and up. The image of the point is point E. What point is the pre-image of point E?
A
B
C
D

Answer 1

To find the pre-image of point E (7, 7) after a translation, we need to determine the amount the point was translated to the right and up.

Let's assume that the translation was by \( (x, y) \) units to the right and up, respectively.

If the pre-image is point P and it is translated to point E, we can express this transformation as:
\[
E = P + (x, y)
\]
Given \( E(7, 7) \), we can express this as:
\[
(7, 7) = (P_x + x, P_y + y)
\]
where \( P_x \) and \( P_y \) are the coordinates of the pre-image point P.

We know the coordinates of other points (A, B, C, D), and we can find if any of them match the movement required to arrive at point E.

1. **From Point A (2, 7)**:
\[
E = (2 + x, 7 + y) \Rightarrow x = 7 - 2 = 5, \quad y = 7 - 7 = 0
\]
Translated Point A would be (2 + 5, 7 + 0) = (7, 7) = E.

2. **From Point B (2, 1)**:
\[
E = (2 + x, 1 + y) \Rightarrow x = 7 - 2 = 5, \quad y = 7 - 1 = 6
\]
This means it would have been translated 5 units right and 6 units up.

3. **From Point C (9, 8)**:
\[
E = (9 + x, 8 + y) \Rightarrow x = 7 - 9 = -2, \quad y = 7 - 8 = -1
\]
This means it would have been translated -2 units left and -1 unit down.

4. **From Point D (8, 3)**:
\[
E = (8 + x, 3 + y) \Rightarrow x = 7 - 8 = -1, \quad y = 7 - 3 = 4
\]
This means it has been translated -1 unit left and 4 units up.

The only point that translates directly to point E (7, 7) is point A (2, 7). It has been moved 5 units to the right and 0 units up.

Thus, the pre-image of point E is point A (2, 7).