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

Answer 1

To find the coordinates of point P’ after translating point P (7, 4) 4 units to the left and 8 units down, we can follow these steps:

1. **Translate 4 units to the left**:
- From x = 7, moving left by 4 units gives us:
\[
7 - 4 = 3
\]

2. **Translate 8 units down**:
- From y = 4, moving down by 8 units gives us:
\[
4 - 8 = -4
\]

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

Next, we need to calculate the distance between points P (7, 4) and P’ (3, -4). We can use the distance formula:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
where (x₁, y₁) = (7, 4) and (x₂, y₂) = (3, -4).

Calculating step-by-step:

1. Calculate \(x_2 - x_1\):
\[
3 - 7 = -4
\]

2. Calculate \(y_2 - y_1\):
\[
-4 - 4 = -8
\]

3. Square the differences:
\[
(-4)^2 = 16 \quad \text{and} \quad (-8)^2 = 64
\]

4. Sum the squared differences:
\[
16 + 64 = 80
\]

5. Take the square root:
\[
d = \sqrt{80} = \sqrt{16 \times 5} = 4\sqrt{5}
\]

To find a numerical approximation for \( \sqrt{5} \), we can use the fact that \( \sqrt{5} \) is approximately 2.236:
\[
d \approx 4 \times 2.236 \approx 8.944
\]

This is approximately 8.9, which, rounding to one decimal place, is 8.9.

Thus, the answer is:
**A. 8.9**