Let's walk through the problem step-by-step to understand where Heather might have gone wrong.
We are given two points:
- Point [tex]\( R(-3, -4) \)[/tex]
- Point [tex]\( S(5, 7) \)[/tex]
The formula to calculate the distance between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
1. Calculate the Differences:
[tex]\[
\Delta x = x_2 - x_1 = 5 - (-3) = 5 + 3 = 8
\][/tex]
[tex]\[
\Delta y = y_2 - y_1 = 7 - (-4) = 7 + 4 = 11
\][/tex]
2. Square the Differences:
[tex]\[
(\Delta x)^2 = 8^2 = 64
\][/tex]
[tex]\[
(\Delta y)^2 = 11^2 = 121
\][/tex]
3. Add the Squares:
[tex]\[
\text{Sum of squares} = 64 + 121 = 185
\][/tex]
4. Calculate the Distance:
[tex]\[
d = \sqrt{185} \approx 13.60
\][/tex]
Now, let's analyze Heather's calculations:
[tex]\[
RS = \sqrt{((-4)-(-3))^2 + (7-5)^2}
\][/tex]
[tex]\[
RS = \sqrt{(-1)^2 + (2)^2}
\][/tex]
[tex]\[
RS = \sqrt{1 + 4}
\][/tex]
[tex]\[
RS = \sqrt{5}
\][/tex]
Heather's Calculation:
[tex]\[
RS = \sqrt{5} \approx 2.236
\][/tex]
Compare Heather's calculation with the correct calculation.
Heather substituted [tex]\((x_2, y_1)\)[/tex] and [tex]\((y_2 - y_1)\)[/tex] as follows:
[tex]\[
\Delta x = -4 - (-3) = -1
\][/tex]
[tex]\[
\Delta y = 7 - 5 = 2
\][/tex]
While the correct differences should be:
[tex]\[
x_2 - x_1 = 5 - (-3) = 8
\][/tex]
[tex]\[
y_2 - y_1 = 7 - (-4) = 11
\][/tex]
When Heather calculated the distance, she used the wrong differences. Therefore, the error lies in how she substituted the coordinates into the distance formula.
Thus, the correct answer is:
A. She substituted incorrectly into the distance formula.
