Heather's work to find the distance between the points \( R(-3, -4) \) and \( S(5, 7) \) can be evaluated using the distance formula. The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
[tex]\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Let's apply the coordinates for points \( R \) and \( S \) to this formula:
1. For point \( R(-3, -4) \) and point \( S(5, 7) \), we have:
[tex]\[
x_1 = -3, \quad y_1 = -4, \quad x_2 = 5, \quad y_2 = 7
\][/tex]
2. Substituting these values into the distance formula, we get:
[tex]\[
d = \sqrt{(5 - (-3))^2 + (7 - (-4))^2}
\][/tex]
3. Simplify the expressions inside the parentheses:
[tex]\[
d = \sqrt{(5 + 3)^2 + (7 + 4)^2}
\][/tex]
[tex]\[
d = \sqrt{8^2 + 11^2}
\][/tex]
4. Continue to simplify by calculating the squares:
[tex]\[
d = \sqrt{64 + 121}
\][/tex]
5. Add the squared terms together:
[tex]\[
d = \sqrt{185}
\][/tex]
Comparing this to Heather's steps:
- She substituted into the distance formula as
[tex]\[
\sqrt{((-4) - (-3))^2 + (7 - 5)^2}
\][/tex]
This should have been
[tex]\[
\sqrt{(5 - (-3))^2 + (7 - (-4))^2}
\][/tex]
Thus, she confused the subtraction terms and made a substitution error.
Therefore, the correct answer is:
A. She substituted incorrectly into the distance formula.
