Answer :
To determine the correctness of Heather’s calculation, let's analyze her application of the distance formula step-by-step.
The distance formula to find 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]
Given points [tex]\(R(-3,-4)\)[/tex] and [tex]\(S(5,7)\)[/tex], we substitute [tex]\(x_1 = -3\)[/tex], [tex]\(y_1 = -4\)[/tex], [tex]\(x_2 = 5\)[/tex], and [tex]\(y_2 = 7\)[/tex] into the formula.
1. Calculate [tex]\((x_2 - x_1)\)[/tex]:
[tex]\[ x_2 - x_1 = 5 - (-3) = 5 + 3 = 8 \][/tex]
2. Calculate [tex]\((y_2 - y_1)\)[/tex]:
[tex]\[ y_2 - y_1 = 7 - (-4) = 7 + 4 = 11 \][/tex]
3. Square the differences:
[tex]\[ (x_2 - x_1)^2 = 8^2 = 64 \][/tex]
[tex]\[ (y_2 - y_1)^2 = 11^2 = 121 \][/tex]
4. Add the squared differences:
[tex]\[ (x_2 - x_1)^2 + (y_2 - y_1)^2 = 64 + 121 = 185 \][/tex]
5. Take the square root to find the distance:
[tex]\[ d = \sqrt{185} \approx 13.601470508735444 \][/tex]
Now, let's analyze Heather’s calculation:
[tex]\[ R S = \sqrt{((-4)-(-3))^2+(7-5)^2} \][/tex]
She found:
[tex]\[ (-4) - (-3) = -1 \quad \text{and} \quad 7 - 5 = 2 \][/tex]
Thus, her equation becomes:
[tex]\[ RS = \sqrt{(-1)^2 + 2^2} = \sqrt{1 + 4} = \sqrt{5} \approx 2.23606797749979 \][/tex]
Heather’s calculation gave the distance as [tex]\(\sqrt{5}\)[/tex] which is incorrect because she incorrectly substituted into the distance formula.
Correct substitution would be:
[tex]\[ RS = \sqrt{(5 - (-3))^2 + (7 - (-4))^2} = \sqrt{8^2 + 11^2} = \sqrt{64 + 121} = \sqrt{185} \approx 13.601470508735444 \][/tex]
Therefore, Heather's error is outlined by the correct choice:
A. She substituted incorrectly into the distance formula.
The distance formula to find 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]
Given points [tex]\(R(-3,-4)\)[/tex] and [tex]\(S(5,7)\)[/tex], we substitute [tex]\(x_1 = -3\)[/tex], [tex]\(y_1 = -4\)[/tex], [tex]\(x_2 = 5\)[/tex], and [tex]\(y_2 = 7\)[/tex] into the formula.
1. Calculate [tex]\((x_2 - x_1)\)[/tex]:
[tex]\[ x_2 - x_1 = 5 - (-3) = 5 + 3 = 8 \][/tex]
2. Calculate [tex]\((y_2 - y_1)\)[/tex]:
[tex]\[ y_2 - y_1 = 7 - (-4) = 7 + 4 = 11 \][/tex]
3. Square the differences:
[tex]\[ (x_2 - x_1)^2 = 8^2 = 64 \][/tex]
[tex]\[ (y_2 - y_1)^2 = 11^2 = 121 \][/tex]
4. Add the squared differences:
[tex]\[ (x_2 - x_1)^2 + (y_2 - y_1)^2 = 64 + 121 = 185 \][/tex]
5. Take the square root to find the distance:
[tex]\[ d = \sqrt{185} \approx 13.601470508735444 \][/tex]
Now, let's analyze Heather’s calculation:
[tex]\[ R S = \sqrt{((-4)-(-3))^2+(7-5)^2} \][/tex]
She found:
[tex]\[ (-4) - (-3) = -1 \quad \text{and} \quad 7 - 5 = 2 \][/tex]
Thus, her equation becomes:
[tex]\[ RS = \sqrt{(-1)^2 + 2^2} = \sqrt{1 + 4} = \sqrt{5} \approx 2.23606797749979 \][/tex]
Heather’s calculation gave the distance as [tex]\(\sqrt{5}\)[/tex] which is incorrect because she incorrectly substituted into the distance formula.
Correct substitution would be:
[tex]\[ RS = \sqrt{(5 - (-3))^2 + (7 - (-4))^2} = \sqrt{8^2 + 11^2} = \sqrt{64 + 121} = \sqrt{185} \approx 13.601470508735444 \][/tex]
Therefore, Heather's error is outlined by the correct choice:
A. She substituted incorrectly into the distance formula.
