To determine the correct distance between the points [tex]\( R(-3, -4) \)[/tex] and [tex]\( S(5, 7) \)[/tex], we use the distance formula, which is given by:
[tex]\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Here, the coordinates are:
[tex]\[
(x_1, y_1) = (-3, -4)
\][/tex]
[tex]\[
(x_2, y_2) = (5, 7)
\][/tex]
Now let's plug the values into the formula step-by-step:
1. First, calculate [tex]\( (x_2 - x_1) \)[/tex]:
[tex]\[
x_2 - x_1 = 5 - (-3) = 5 + 3 = 8
\][/tex]
2. Then, calculate [tex]\( (y_2 - y_1) \)[/tex]:
[tex]\[
y_2 - y_1 = 7 - (-4) = 7 + 4 = 11
\][/tex]
3. Square both results:
[tex]\[
(8)^2 = 64
\][/tex]
[tex]\[
(11)^2 = 121
\][/tex]
4. Add the squares:
[tex]\[
64 + 121 = 185
\][/tex]
5. Finally, take the square root:
[tex]\[
d = \sqrt{185} \approx 13.601
\][/tex]
Now, let's review Heather's steps:
[tex]\[
\begin{aligned}
R S & = \sqrt{((-4) - (-3))^2 + (7 - 5)^2} \\
& = \sqrt{(-1)^2 + (2)^2} \\
& = \sqrt{1 + 4} \\
& = \sqrt{5} \approx 2.236
\end{aligned}
\][/tex]
Heather incorrectly calculated the differences in the coordinates:
- She calculated [tex]\( (-4) - (-3) = -1 \)[/tex] instead of [tex]\( 7 - (-4) \)[/tex].
- She calculated [tex]\( 7 - 5 = 2 \)[/tex] instead of [tex]\( 5 - (-3) \)[/tex].
Therefore, the error is:
- A. She substituted incorrectly into the distance formula.
