To determine whether Heather made an error, we need to correctly apply the distance formula for two points in a coordinate plane. The distance formula for two points [tex]\( R(x_1, y_1) \)[/tex] and [tex]\( S(x_2, y_2) \)[/tex] is given by:
[tex]\[ RS = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Here, the points given are [tex]\( R(-3, -4) \)[/tex] and [tex]\( S(5, 7) \)[/tex]. We will substitute these coordinates into the formula:
1. Calculate the differences in the x-coordinates and the y-coordinates:
[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]
2. Substitute these differences into the distance formula:
[tex]\[
RS = \sqrt{(\Delta x)^2 + (\Delta y)^2} = \sqrt{(8)^2 + (11)^2}
\][/tex]
[tex]\[
RS = \sqrt{64 + 121}
\][/tex]
[tex]\[
RS = \sqrt{185}
\][/tex]
Comparing this with Heather's work:
[tex]\[
RS = \sqrt{((-4) - (-3))^2 + (7 - 5)^2} = \sqrt{(-1)^2 + (2)^2} = \sqrt{1 + 4} = \sqrt{5}
\][/tex]
We see that Heather's approach has the following errors:
- Incorrectly calculated differences in coordinates.
- Heather used [tex]\((-4) - (-3)\)[/tex] for [tex]\(x\)[/tex]-coordinate, which simplifies to [tex]\(-1\)[/tex].
- Heather used [tex]\(7 - 5\)[/tex] for the [tex]\(y\)[/tex]-coordinate, which results in [tex]\(2\)[/tex].
Both of these differences are incorrect according to the distance formula.
Thus, the correct choice that describes Heather's error is:
A. She substituted incorrectly into the distance formula.
