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?

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 perform the following calculations:

1. Translate point P 4 units to the left:
\( 7 - 4 = 3 \)
2. Translate point P 8 units down:
\( 4 - 8 = -4 \)

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

Now, we can calculate the distance between points P (7, 4) and P' (3, -4) using the distance formula:

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

Substituting the coordinates of P and P':

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

Now we find the numerical value of \(4\sqrt{5}\). Since \(\sqrt{5} \approx 2.236\):

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

The measurement closest to the distance between point P and point P' is approximately **8.9 units**.