Question

Point A is located at (−5, 2) on a coordinate grid. Point A is translated 8 units to the right and 3 units up to create point A'. Which measurement is closest to the distance between point A and point A' in units?
Responses
A 8.1 units8.1 units
B 8.5 units8.5 units
C 8.9 units8.9 units
D 9.4 units

Answer 1

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:

1. **Translation to the right**: To move 8 units to the right, we add 8 to the x-coordinate of point A.
\[
x' = -5 + 8 = 3
\]

2. **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**.