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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]\[
\begin{aligned}
RS & = \sqrt{((-4)-(-3))^2+(7-5)^2} \\
& = \sqrt{(-1)^2+(2)^2} \\
& = \sqrt{1+4} \\
& = \sqrt{5}
\end{aligned}
\][/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
To determine the correct distance between the points [tex]\( R(-3, -4) \)[/tex] and [tex]\( S(5, 7) \)[/tex], we use the distance formula, which is given by:

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

Here, the coordinates are:
[tex]\[ (x_1, y_1) = (-3, -4) \][/tex]
[tex]\[ (x_2, y_2) = (5, 7) \][/tex]

Now let's plug the values into the formula step-by-step:

1. First, calculate [tex]\( (x_2 - x_1) \)[/tex]:
[tex]\[ x_2 - x_1 = 5 - (-3) = 5 + 3 = 8 \][/tex]

2. Then, calculate [tex]\( (y_2 - y_1) \)[/tex]:
[tex]\[ y_2 - y_1 = 7 - (-4) = 7 + 4 = 11 \][/tex]

3. Square both results:
[tex]\[ (8)^2 = 64 \][/tex]
[tex]\[ (11)^2 = 121 \][/tex]

4. Add the squares:
[tex]\[ 64 + 121 = 185 \][/tex]

5. Finally, take the square root:
[tex]\[ d = \sqrt{185} \approx 13.601 \][/tex]

Now, let's review Heather's steps:
[tex]\[ \begin{aligned} R S & = \sqrt{((-4) - (-3))^2 + (7 - 5)^2} \\ & = \sqrt{(-1)^2 + (2)^2} \\ & = \sqrt{1 + 4} \\ & = \sqrt{5} \approx 2.236 \end{aligned} \][/tex]

Heather incorrectly calculated the differences in the coordinates:
- She calculated [tex]\( (-4) - (-3) = -1 \)[/tex] instead of [tex]\( 7 - (-4) \)[/tex].
- She calculated [tex]\( 7 - 5 = 2 \)[/tex] instead of [tex]\( 5 - (-3) \)[/tex].

Therefore, the error is:
- A. She substituted incorrectly into the distance formula.

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