Answer :
Heather's work contains an error in her substitution into the distance formula. Let's identify where she went wrong and determine the correct approach.
Given the points [tex]\( R(-3, -4) \)[/tex] and [tex]\( S(5, 7) \)[/tex], it's essential to use the distance formula correctly:
[tex]\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
For our specific points:
[tex]\( x_1 = -3 \)[/tex], [tex]\( y_1 = -4 \)[/tex], [tex]\( x_2 = 5 \)[/tex], and [tex]\( y_2 = 7 \)[/tex]
Let's execute the steps carefully:
1. Calculate the difference in the x-coordinates:
[tex]\[ x_2 - x_1 = 5 - (-3) = 5 + 3 = 8 \][/tex]
2. Calculate the difference in the y-coordinates:
[tex]\[ y_2 - y_1 = 7 - (-4) = 7 + 4 = 11 \][/tex]
3. Square both differences:
[tex]\[ 8^2 = 64 \][/tex]
[tex]\[ 11^2 = 121 \][/tex]
4. Sum the squares of the differences:
[tex]\[ 64 + 121 = 185 \][/tex]
5. Take the square root of the result:
[tex]\[ \sqrt{185} \approx 13.601470508735444 \][/tex]
Comparing this with Heather's solution, we see that Heather initially made a mistake in her point coordinates and their differences:
[tex]\[ \begin{aligned} & \sqrt{((-4)-(-3))^2+(7-5)^2} \\ & = \sqrt{((-4)+3)^2+(7-5)^2} \\ & = \sqrt{(-1)^2 + (2)^2} \\ & = \sqrt{1 + 4} \\ & = \sqrt{5} \end{aligned} \][/tex]
This derivation is incorrect as it's derived from improper substitution values.
Therefore, the correct response is:
A. She substituted incorrectly into the distance formula.
Given the points [tex]\( R(-3, -4) \)[/tex] and [tex]\( S(5, 7) \)[/tex], it's essential to use the distance formula correctly:
[tex]\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
For our specific points:
[tex]\( x_1 = -3 \)[/tex], [tex]\( y_1 = -4 \)[/tex], [tex]\( x_2 = 5 \)[/tex], and [tex]\( y_2 = 7 \)[/tex]
Let's execute the steps carefully:
1. Calculate the difference in the x-coordinates:
[tex]\[ x_2 - x_1 = 5 - (-3) = 5 + 3 = 8 \][/tex]
2. Calculate the difference in the y-coordinates:
[tex]\[ y_2 - y_1 = 7 - (-4) = 7 + 4 = 11 \][/tex]
3. Square both differences:
[tex]\[ 8^2 = 64 \][/tex]
[tex]\[ 11^2 = 121 \][/tex]
4. Sum the squares of the differences:
[tex]\[ 64 + 121 = 185 \][/tex]
5. Take the square root of the result:
[tex]\[ \sqrt{185} \approx 13.601470508735444 \][/tex]
Comparing this with Heather's solution, we see that Heather initially made a mistake in her point coordinates and their differences:
[tex]\[ \begin{aligned} & \sqrt{((-4)-(-3))^2+(7-5)^2} \\ & = \sqrt{((-4)+3)^2+(7-5)^2} \\ & = \sqrt{(-1)^2 + (2)^2} \\ & = \sqrt{1 + 4} \\ & = \sqrt{5} \end{aligned} \][/tex]
This derivation is incorrect as it's derived from improper substitution values.
Therefore, the correct response is:
A. She substituted incorrectly into the distance formula.
