Answer :
Heather's solution for finding the distance between the points \( R(-3, -4) \) and \( S(5, 7) \) uses the distance formula correctly. Let’s review each step carefully to identify any potential mistakes:
1. The distance formula to find the distance \( d \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
2. Substituting the given points \( R(-3, -4) \) and \( S(5, 7) \):
\( x_1 = -3 \)
\( y_1 = -4 \)
\( x_2 = 5 \)
\( y_2 = 7 \)
The correct substitution into the distance formula is:
[tex]\[ RS = \sqrt{(5 - (-3))^2 + (7 - (-4))^2} \][/tex]
3. Calculating the differences:
[tex]\[ 5 - (-3) = 5 + 3 = 8 \][/tex]
[tex]\[ 7 - (-4) = 7 + 4 = 11 \][/tex]
4. Substitute these results back into the formula:
[tex]\[ RS = \sqrt{8^2 + 11^2} \][/tex]
5. Simplify inside the radical:
[tex]\[ 8^2 = 64 \][/tex]
[tex]\[ 11^2 = 121 \][/tex]
[tex]\[ 64 + 121 = 185 \][/tex]
[tex]\[ RS = \sqrt{185} \][/tex]
Heather’s original work shows the following steps:
[tex]\[ \begin{aligned} RS & =\sqrt{((-4)-(-3))^2+(7-5)^2} \\ & =\sqrt{(-1)^2+(2)^2} \\ & =\sqrt{1+4} \\ & =\sqrt{5} \end{aligned} \][/tex]
From the correct calculations and seeing Heather's work, it is clear that Heather made a mistake in her initial substitution into the distance formula. She swapped the coordinates and used the wrong formula:
- She incorrectly calculated \( x_2 - x_1 \) as \( (-4) - (-3) \) instead of \( 5 - (-3) \).
- She incorrectly calculated \( y_2 - y_1 \) as \( 7 - 5 \) instead of \( 7 - (-4) \).
Thus, the answer is:
A. She substituted incorrectly into the distance formula.
1. The distance formula to find the distance \( d \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
2. Substituting the given points \( R(-3, -4) \) and \( S(5, 7) \):
\( x_1 = -3 \)
\( y_1 = -4 \)
\( x_2 = 5 \)
\( y_2 = 7 \)
The correct substitution into the distance formula is:
[tex]\[ RS = \sqrt{(5 - (-3))^2 + (7 - (-4))^2} \][/tex]
3. Calculating the differences:
[tex]\[ 5 - (-3) = 5 + 3 = 8 \][/tex]
[tex]\[ 7 - (-4) = 7 + 4 = 11 \][/tex]
4. Substitute these results back into the formula:
[tex]\[ RS = \sqrt{8^2 + 11^2} \][/tex]
5. Simplify inside the radical:
[tex]\[ 8^2 = 64 \][/tex]
[tex]\[ 11^2 = 121 \][/tex]
[tex]\[ 64 + 121 = 185 \][/tex]
[tex]\[ RS = \sqrt{185} \][/tex]
Heather’s original work shows the following steps:
[tex]\[ \begin{aligned} RS & =\sqrt{((-4)-(-3))^2+(7-5)^2} \\ & =\sqrt{(-1)^2+(2)^2} \\ & =\sqrt{1+4} \\ & =\sqrt{5} \end{aligned} \][/tex]
From the correct calculations and seeing Heather's work, it is clear that Heather made a mistake in her initial substitution into the distance formula. She swapped the coordinates and used the wrong formula:
- She incorrectly calculated \( x_2 - x_1 \) as \( (-4) - (-3) \) instead of \( 5 - (-3) \).
- She incorrectly calculated \( y_2 - y_1 \) as \( 7 - 5 \) instead of \( 7 - (-4) \).
Thus, the answer is:
A. She substituted incorrectly into the distance formula.
