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Select the correct answer.

Heather's work to find the distance between two points, [tex]\( A(-3, -4) \)[/tex] and [tex]\( S(5, 7) \)[/tex], 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
Let's analyze Heather's work and identify any potential errors.

Heather's calculation is given by:

1. She starts by using the distance formula:

[tex]\( \sqrt{((-4) - (-3))^2 + (7 - 5)^2} \)[/tex]

2. Simplifies the expressions inside the square root:

[tex]\( \sqrt{(-1)^2 + (2)^2} \)[/tex]

3. Squares the simplified values:

[tex]\( \sqrt{1 + 4} \)[/tex]

4. Adds the squares:

[tex]\( \sqrt{5} \)[/tex]

Her final answer is [tex]\( \sqrt{5} \)[/tex].

To find the correct distance between points [tex]\( A(-3,-4) \)[/tex] and [tex]\( S(5,7) \)[/tex], we use the correct distance formula, which is:

[tex]\[ \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]

Substituting the coordinates of points [tex]\( A \)[/tex] and [tex]\( S \)[/tex]:

[tex]\[ \sqrt{(5 - (-3))^2 + (7 - (-4))^2} \][/tex]

This simplifies to:

[tex]\[ \sqrt{(5 + 3)^2 + (7 + 4)^2} \][/tex]

Further simplification gives:

[tex]\[ \sqrt{8^2 + 11^2} \][/tex]

Calculating the squares:

[tex]\[ \sqrt{64 + 121} \][/tex]

Adding the squares:

[tex]\[ \sqrt{185} \][/tex]

The correct distance between the points [tex]\( A \)[/tex] and [tex]\( S \)[/tex] is [tex]\( \sqrt{185} \)[/tex], which is approximately 13.601.

Therefore, Heather's result of [tex]\( \sqrt{5} \)[/tex] is significantly different from the correct distance. The source of her error is apparent when we see that she incorrectly substituted into the distance formula.

The correct identification of Heather's error is:
A. She substituted incorrectly into the distance formula.

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