To find the distance between two points [tex]\( R(-3, -4) \)[/tex] and [tex]\( S(5, 7) \)[/tex], we use the distance formula:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Substitute the coordinates of points [tex]\( R \)[/tex] and [tex]\( S \)[/tex]:
[tex]\[ d = \sqrt{(5 - (-3))^2 + (7 - (-4))^2} \][/tex]
[tex]\[ d = \sqrt{(5 + 3)^2 + (7 + 4)^2} \][/tex]
[tex]\[ d = \sqrt{8^2 + 11^2} \][/tex]
[tex]\[ d = \sqrt{64 + 121} \][/tex]
[tex]\[ d = \sqrt{185} \][/tex]
Thus, the correct calculation gives us a distance of [tex]\( \sqrt{185} \)[/tex].
Let's review Heather's work:
[tex]\[
\begin{aligned}
RS & = \sqrt{((-4) - (-3))^2 + (7 - 5)^2} \\
& = \sqrt{(-1)^2 + (2)^2} \\
& = \sqrt{1 + 4} \\
& = \sqrt{5}
\end{aligned}
\][/tex]
Heather substituted [tex]\((-4) - (-3)\)[/tex] and [tex]\(7 - 5\)[/tex] into the distance formula incorrectly. She used the [tex]\( y \)[/tex]-coordinates of point [tex]\( R \)[/tex] and point [tex]\( S \)[/tex] as one pair and the [tex]\( x \)[/tex]-coordinates also as a pair but in the wrong manner.
The error she made is option:
A. She substituted incorrectly into the distance formula.
