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Select the correct answer.

Heather's work to find the distance between two points, [tex]R(-3, -4)[/tex] and [tex]S(5, 7)[/tex], is shown:

[tex]
\begin{aligned}
R S & =\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
To find the distance between two points [tex]\( R(-3, -4) \)[/tex] and [tex]\( S(5, 7) \)[/tex], we use the distance formula:

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

Substitute the coordinates of points [tex]\( R \)[/tex] and [tex]\( S \)[/tex]:

[tex]\[ d = \sqrt{(5 - (-3))^2 + (7 - (-4))^2} \][/tex]
[tex]\[ d = \sqrt{(5 + 3)^2 + (7 + 4)^2} \][/tex]
[tex]\[ d = \sqrt{8^2 + 11^2} \][/tex]
[tex]\[ d = \sqrt{64 + 121} \][/tex]
[tex]\[ d = \sqrt{185} \][/tex]

Thus, the correct calculation gives us a distance of [tex]\( \sqrt{185} \)[/tex].

Let's review Heather's work:

[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 substituted [tex]\((-4) - (-3)\)[/tex] and [tex]\(7 - 5\)[/tex] into the distance formula incorrectly. She used the [tex]\( y \)[/tex]-coordinates of point [tex]\( R \)[/tex] and point [tex]\( S \)[/tex] as one pair and the [tex]\( x \)[/tex]-coordinates also as a pair but in the wrong manner.

The error she made is option:

A. She substituted incorrectly into the distance formula.

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