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Select the correct answer.

Heather's work to find the distance between two points, \( R(-3,-4) \) and \( S(5,7) \), is shown:
[tex]\[
\begin{aligned}
RS & = \sqrt{((-4)-(-3))^2 + (7-5)^2} \\
& = \sqrt{(-1)^2 + (2)^2} \\
& = \sqrt{1 + 4} \\
& = \sqrt{5}
\end{aligned}
\][/tex]

What error, if any, did Heather make?

A. She substituted incorrectly into the distance formula.
B. She subtracted the coordinates instead of adding them.
C. She made a sign error when simplifying inside the radical.
D. She made no errors.



Answer :

GinnyAnswer
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.

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