To determine whether Heather made any errors in her calculation, let's carefully walk through the correct steps to find the distance between two points, \( R(-3, -4) \) and \( S(5, 7) \), using the distance formula. The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
[tex]\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Substituting the given coordinates \((x_1 = -3, y_1 = -4)\) and \((x_2 = 5, y_2 = 7)\):
1. Calculate \( x_2 - x_1 \) and \( y_2 - y_1 \):
[tex]\[
x_2 - x_1 = 5 - (-3) = 5 + 3 = 8
\][/tex]
[tex]\[
y_2 - y_1 = 7 - (-4) = 7 + 4 = 11
\][/tex]
2. Plug these differences back into the formula:
[tex]\[
\text{Distance} = \sqrt{(8)^2 + (11)^2}
\][/tex]
3. Square the differences:
[tex]\[
8^2 = 64
\][/tex]
[tex]\[
11^2 = 121
\][/tex]
4. Add the squares and take the square root:
[tex]\[
\text{Distance} = \sqrt{64 + 121} = \sqrt{185}
\][/tex]
[tex]\[
\text{Distance} \approx 13.60
\][/tex]
Now, let's review Heather's work to identify any errors.
Heather's calculations:
[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 computed:
1. [tex]\[ (-4) - (-3) = -4 + 3 = -1 \][/tex]
2. [tex]\[ 7 - 5 = 2 \][/tex]
Therefore:
[tex]\[
RS = \sqrt{(-1)^2 + (2)^2} = \sqrt{1 + 4} = \sqrt{5} \approx 2.24
\][/tex]
Note the discrepancy:
- Correct computation resulted in a distance of approximately 13.60.
- Heather's computation gave a distance of approximately 2.24 (i.e., \(\sqrt{5}\)).
Heather made an error in the subtraction inside the radical:
- Specifically, she simplified \((-4) - (-3)\) to \(-1\) instead of properly considering the coordinates \((x_1 = -3)\) and \((x_2 = 5)\).
This leads us to conclude:
[tex]\[ \boxed{C. \text{She made a sign error when simplifying inside the radical.}} \][/tex]
