To analyze George's work, let’s solve the problem step-by-step using the distance formula.
The distance formula 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 the points [tex]\((2, 4)\)[/tex] and [tex]\((6, 3)\)[/tex]:
1. Calculate the differences:
[tex]\[
\Delta x = x_2 - x_1 = 6 - 2 = 4
\][/tex]
[tex]\[
\Delta y = y_2 - y_1 = 3 - 4 = -1
\][/tex]
2. Substitute these differences into the distance formula:
[tex]\[
d = \sqrt{(4)^2 + (-1)^2}
\][/tex]
3. Evaluate the squares of these differences:
[tex]\[
d = \sqrt{16 + 1}
\][/tex]
4. Add the results inside the square root:
[tex]\[
d = \sqrt{17}
\][/tex]
Thus, the value of [tex]\(\sqrt{17}\)[/tex] is approximately [tex]\(4.123105625617661\)[/tex].
Let’s now compare these steps with George’s steps:
1. George's first step:
[tex]\[
d = \sqrt{(6-2)^2 + (3-4)^2}
\][/tex]
- Matches the differences calculated correctly.
2. George's second step:
[tex]\[
d = \sqrt{(4)^2 + (-1)^2}
\][/tex]
- Correctly substituted the differences [tex]\(\Delta x = 4\)[/tex] and [tex]\(\Delta y = -1\)[/tex].
3. George's third step:
[tex]\[
d = \sqrt{16 + 1}
\][/tex]
- Correctly evaluated the squares.
4. George's fourth step:
[tex]\[
d = \sqrt{17}
\][/tex]
- Correctly added the values inside the square root.
Hence, from the above analysis, it is clear that George has followed the proper steps and calculations correctly. Thus, the answer choices indicate that:
- Yes, he is correct.
George made no mistakes in substitution, order of operations, or evaluating powers. Therefore, his calculation is accurate, and he is correct.
