Heather is trying to find the distance between two points, [tex]\( R(-3,-4) \)[/tex] and [tex]\( S(5,7) \)[/tex], using the distance formula. The distance formula is:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Let's explicitly compare Heather's steps to the correct steps:
1. Identify Coordinates:
- [tex]\( R(x_1, y_1) = (-3, -4) \)[/tex]
- [tex]\( S(x_2, y_2) = (5, 7) \)[/tex]
2. Substitute the Coordinates into the Distance Formula:
- Correct Calculation:
[tex]\[ d = \sqrt{(5 - (-3))^2 + (7 - (-4))^2} \][/tex]
- Simplify inside the parentheses:
[tex]\[ d = \sqrt{(5 + 3)^2 + (7 + 4)^2} \][/tex]
- Continue simplifying:
[tex]\[ d = \sqrt{8^2 + 11^2} \][/tex]
[tex]\[ d = \sqrt{64 + 121} \][/tex]
[tex]\[ d = \sqrt{185} \][/tex]
[tex]\[ d \approx 13.601 \][/tex]
3. Heather's Calculation:
- Heather wrote:
[tex]\[ d = \sqrt{((-4) - (-3))^2 + (7 - 5)^2} \][/tex]
- Simplify inside the parentheses:
[tex]\[ d = \sqrt{(-1)^2 + 2^2} \][/tex]
[tex]\[ d = \sqrt{1 + 4} \][/tex]
[tex]\[ d = \sqrt{5} \][/tex]
[tex]\[ d \approx 2.236 \][/tex]
Error Analysis:
- Heather made her first subtraction incorrectly. According to her calculation, she used:
[tex]\[ (-4) - (-3) = -1 \][/tex]
- The correct x-component calculation should be:
[tex]\[ x_2 - x_1 = 5 - (-3) = 8 \][/tex]
- Heather's subtraction for the y-coordinates was correct:
[tex]\[ y_2 - y_1 = 7 - 5 = 2 \][/tex]
Given these points, we can see Heather's incorrect handling specifically when calculating the x-component of the distance. Hence, she made an error in her substitution step. Therefore, the correct answer to the question is:
A. She substituted incorrectly into the distance formula.
