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