To determine the distance between two points [tex]\( R(-3, -4) \)[/tex] and [tex]\( S(5, 7) \)[/tex], we use the distance formula:
[tex]\[
RS = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
For points [tex]\( R(-3, -4) \)[/tex] and [tex]\( S(5, 7) \)[/tex], the coordinates are:
[tex]\( x_1 = -3\)[/tex], [tex]\( y_1 = -4\)[/tex], [tex]\( x_2 = 5 \)[/tex], and [tex]\( y_2 = 7 \)[/tex].
Substitute these coordinates into the distance formula:
[tex]\[
RS = \sqrt{(5 - (-3))^2 + (7 - (-4))^2}
\][/tex]
Simplify inside the parentheses:
[tex]\[
RS = \sqrt{(5 + 3)^2 + (7 + 4)^2}
\][/tex]
Further simplifying:
[tex]\[
RS = \sqrt{8^2 + 11^2}
\][/tex]
Calculate the squares:
[tex]\[
RS = \sqrt{64 + 121}
\][/tex]
Add the results inside the radical:
[tex]\[
RS = \sqrt{185}
\][/tex]
So, the correct distance is:
[tex]\[
RS \approx 13.6
\][/tex]
Now, let's compare this with Heather's calculation. Heather's steps were:
[tex]\[
RS = \sqrt{((-4)-(-3))^2 + (7-5)^2}
\][/tex]
[tex]\[
= \sqrt{(-1)^2 + (2)^2}
\][/tex]
[tex]\[
= \sqrt{1 + 4}
\][/tex]
[tex]\[
= \sqrt{5}
\][/tex]
Heather's final result was [tex]\( \sqrt{5} \approx 2.24\)[/tex]. Upon reviewing her steps, we see that:
1. She substituted the coordinates incorrectly:
[tex]\[
((-4)-(-3))^2 \Rightarrow (-4 + 3)^2 = (-1)^2
\][/tex]
[tex]\[
(7-5)^2 = 2^2
\][/tex]
Instead of correctly using:
[tex]\[
(5 - (-3))^2 \Rightarrow (5 + 3)^2 = 8^2
\][/tex]
[tex]\[
(7 - (-4))^2 \Rightarrow (7 + 4)^2 = 11^2
\][/tex]
This indicates that Heather substituted the coordinates incorrectly into the distance formula.
Therefore, the correct answer is:
A. She substituted incorrectly into the distance formula
