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Question
Point P is located at (7, 4) on a coordinate grid. Point P is translated 4 units to the left and 8 units down to create point P’.

Which measurement is closest to the distance between point P and point P’ in units?
Responses
A 8.98.9
B 4040
C 16.216.2
D 2.65

Answers

GPT-4o mini
To find the new coordinates of point P’ after translating point P (7, 4) 4 units to the left and 8 units down, we perform the following calculations:

1. **Translate 4 units to the left:** Subtract 4 from the x-coordinate.
- New x-coordinate: \( 7 - 4 = 3 \)

2. **Translate 8 units down:** Subtract 8 from the y-coordinate.
- New y-coordinate: \( 4 - 8 = -4 \)

Thus, the coordinates of point P’ are (3, -4).

Next, we can calculate the distance between point P (7, 4) and point P’ (3, -4) using the distance formula:

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

Where \((x_1, y_1)\) are the coordinates of point P, and \((x_2, y_2)\) are the coordinates of point P’.

Plugging in the values:

\[
d = \sqrt{(3 - 7)^2 + (-4 - 4)^2}
\]
\[
= \sqrt{(-4)^2 + (-8)^2}
\]
\[
= \sqrt{16 + 64}
\]
\[
= \sqrt{80}
\]
\[
= \sqrt{16 \cdot 5} = 4\sqrt{5}
\]

To find the approximate numerical value of \( 4\sqrt{5} \):

\[
\sqrt{5} \approx 2.236
\]
Thus,

\[
4\sqrt{5} \approx 4 \times 2.236 \approx 8.944
\]

Now we can check the closest option in the provided responses. The closest option to 8.944 is:

**A. 8.9**

So the correct answer is **A. 8.9**.

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