To determine the correct answer, let's carefully review and compare Heather's steps with the correct method for finding the distance between the two points [tex]\(R(-3, -4)\)[/tex] and [tex]\(S(5, 7)\)[/tex].
The distance formula for two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Here, we have:
- Coordinates of point [tex]\( R \)[/tex]: [tex]\( (x_1, y_1) = (-3, -4) \)[/tex]
- Coordinates of point [tex]\( S \)[/tex]: [tex]\( (x_2, y_2) = (5, 7) \)[/tex]
We need to compute the differences in [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
- [tex]\( \Delta x = x_2 - x_1 = 5 - (-3) = 5 + 3 = 8 \)[/tex]
- [tex]\( \Delta y = y_2 - y_1 = 7 - (-4) = 7 + 4 = 11 \)[/tex]
Next, we use these values in the distance formula:
[tex]\[
d = \sqrt{(8)^2 + (11)^2} = \sqrt{64 + 121} = \sqrt{185}
\][/tex]
This simplifies to approximately [tex]\( \sqrt{185} \approx 13.601 \)[/tex].
Now let's compare this with Heather's calculation:
[tex]\[
\begin{aligned}
R S & = \sqrt{((-4) - (-3))^2 + (7 - 5)^2} \\
& = \sqrt{(-1)^2 + (2)^2} \\
& = \sqrt{1 + 4} \\
& = \sqrt{5}
\end{aligned}
\][/tex]
Heather calculated her differences as:
- [tex]\( \Delta x = -4 - (-3) = -4 + 3 = -1 \)[/tex]
- [tex]\( \Delta y = 7 - 5 = 2 \)[/tex]
We see that Heather incorrectly computed [tex]\(\Delta x\)[/tex] and [tex]\(\Delta y\)[/tex].
Here are the errors Heather made:
- For [tex]\(\Delta x\)[/tex], instead of taking the difference between [tex]\(x_2\)[/tex] and [tex]\(x_1\)[/tex] (which should be [tex]\(5 - (-3)\)[/tex]), she used [tex]\(y\)[/tex]-coordinates incorrectly.
- For [tex]\(\Delta y\)[/tex], instead of using the [tex]\( y \)[/tex]-coordinates difference correctly (which should be [tex]\(7 - (-4)\)[/tex]), she used [tex]\(5\)[/tex] instead of [tex]\(-4\)[/tex].
Thus, the correct answer is C: She made a sign error when simplifying inside the radical.
