To find the coordinates of point A' after translating point A (−5, 2) 8 units to the right and 3 units up, we need to follow these steps:
-
Translation to the right: To move 8 units to the right, we add 8 to the x-coordinate of point A. \[ x' = -5 + 8 = 3 \]
-
Translation upwards: To move 3 units up, we add 3 to the y-coordinate of point A. \[ y' = 2 + 3 = 5 \]
Thus, the new coordinates of point A' are (3, 5).
Now, we calculate the distance between point A (−5, 2) and point A' (3, 5) using the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] where \((x_1, y_1)\) = (−5, 2) and \((x_2, y_2)\) = (3, 5).
Substituting the coordinates into the formula: \[ d = \sqrt{(3 - (-5))^2 + (5 - 2)^2} \] \[ = \sqrt{(3 + 5)^2 + (3)^2} \] \[ = \sqrt{8^2 + 3^2} \] \[ = \sqrt{64 + 9} \] \[ = \sqrt{73} \]
Now we calculate \(\sqrt{73}\). The approximate value of \(\sqrt{73}\) is about 8.544.
Looking at the options provided:
- A: 8.1 units
- B: 8.5 units
- C: 8.9 units
- D: 9.4 units
The closest measurement to \(8.544\) is B: 8.5 units.