Question

point ( p(5, -4) ) is the image of point ( p(2, 3) ) under a translation. select the image of ( (6, -2) ) under the same translation.
( \bigcirc (7, -1) )
( \bigcirc (13, -3) )
( \bigcirc (9, -9) )
( \bigcirc (3, 5) )

Explanation:

Step1: Find the translation vector

To find the translation vector, we subtract the coordinates of the original point \( P(2, 3) \) from the coordinates of its image \( P'(5, -4) \).
For the \( x \)-coordinate: \( 5 - 2 = 3 \)
For the \( y \)-coordinate: \( -4 - 3 = -7 \)
So the translation vector is \( (3, -7) \).

Step2: Apply the translation to the point \( (6, -2) \)

To find the image of \( (6, -2) \) under the same translation, we add the translation vector \( (3, -7) \) to the coordinates of \( (6, -2) \).
For the \( x \)-coordinate: \( 6 + 3 = 9 \)? Wait, no, wait. Wait, let's check again. Wait, maybe I made a mistake. Wait, original point \( P(2,3) \), image \( P'(5, -4) \). So the change in \( x \) is \( 5 - 2 = 3 \), change in \( y \) is \( -4 - 3 = -7 \). So the translation is \( (x + 3, y - 7) \). Wait, but when we apply this to \( (6, -2) \):
\( x \)-coordinate: \( 6 + 3 = 9 \)? But the options don't have \( (9, -9) \) as correct? Wait, no, wait, maybe I messed up. Wait, let's recalculate. Wait, \( P(2,3) \) to \( P'(5, -4) \). So \( x \): \( 5 - 2 = 3 \) (so we add 3 to \( x \)). \( y \): \( -4 - 3 = -7 \) (so we add -7 to \( y \), i.e., subtract 7). So for the point \( (6, -2) \):
\( x \): \( 6 + 3 = 9 \)
\( y \): \( -2 + (-7) = -9 \)
Wait, but \( (9, -9) \) is an option. But let's check the options again. Wait, the options are \( (7, -1) \), \( (13, -3) \), \( (9, -9) \), \( (3, 5) \). Wait, but maybe I made a mistake in the translation. Wait, maybe the translation is \( (x + 3, y - 7) \), so \( (6 + 3, -2 - 7) = (9, -9) \). But let's check again. Wait, original point \( P(2,3) \), image \( P'(5, -4) \). So the translation vector is \( (5 - 2, -4 - 3) = (3, -7) \). So applying this to \( (6, -2) \): \( (6 + 3, -2 + (-7)) = (9, -9) \). But let's check the options. Wait, maybe I made a mistake. Wait, let's check the options again. Wait, the first option is \( (7, -1) \), which would be if the translation was \( (1, 1) \), which is not. Wait, maybe I messed up the direction. Wait, maybe the translation is from \( P' \) to \( P \)? No, the problem says \( P' \) is the image of \( P \) under translation, so \( P \) is original, \( P' \) is image. So translation is \( P' = P + (a, b) \), so \( (a, b) = P' - P = (5 - 2, -4 - 3) = (3, -7) \). So applying to \( (6, -2) \): \( (6 + 3, -2 - 7) = (9, -9) \), which is option C. Wait, but let's check the options again. The options are:

1. \( (7, -1) \)
2. \( (13, -3) \)
3. \( (9, -9) \)
4. \( (3, 5) \)

So \( (9, -9) \) is option 3. But wait, maybe I made a mistake. Wait, let's check the translation again. Wait, \( P(2,3) \) to \( P'(5, -4) \). So \( x \) goes from 2 to 5: that's +3. \( y \) goes from 3 to -4: that's -7. So the translation is \( (x + 3, y - 7) \). So for \( (6, -2) \):

\( x \): 6 + 3 = 9

\( y \): -2 - 7 = -9

So the image is \( (9, -9) \), which is option 3.

Wait, but let's check the options again. The third option is \( (9, -9) \), so that's the answer.

Answer:

\( (9, -9) \) (corresponding to the option with text " (9, −9) ")