To determine which substitutions are incorrect, let's first recall the distance formula for two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex]:
[tex]\[
\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Given the points [tex]\((12, 0)\)[/tex] and [tex]\((-1, 4)\)[/tex], we can substitute these points into the distance formula to find the correct expression for the distance between them.
### Correct substitution:
Substitute the given points into the distance formula:
[tex]\[
\sqrt{((-1) - 12)^2 + (4 - 0)^2} = \sqrt{(-13)^2 + 4^2} = \sqrt{169 + 16} = \sqrt{185}
\][/tex]
Alternatively, another correct form is:
[tex]\[
\sqrt{(12 - (-1))^2 + (0 - 4)^2} = \sqrt{(12 + 1)^2 + (-4)^2} = \sqrt{13^2 + (-4)^2} = \sqrt{169 + 16} = \sqrt{185}
\][/tex]
### Check the given options:
1. [tex]\(\sqrt{(12 - 4)^2 + (0 - (-1))^2}\)[/tex]
Simplify this expression:
[tex]\[
\sqrt{(8)^2 + (1)^2} = \sqrt{64 + 1} = \sqrt{65}
\][/tex]
This is incorrect because it does not yield the same value as [tex]\(\sqrt{185}\)[/tex].
2. [tex]\(\sqrt{(-1 - 12)^2 + (4 - 0)^2}\)[/tex]
Simplify this expression:
[tex]\[
\sqrt{(-13)^2 + 4^2} = \sqrt{169 + 16} = \sqrt{185}
\][/tex]
This is correct as it matches the calculated distance [tex]\(\sqrt{185}\)[/tex].
3. [tex]\(\sqrt{(12 - (-1))^2 + (0 - 4)^2}\)[/tex]
Simplify this expression:
[tex]\[
\sqrt{(12 + 1)^2 + (-4)^2} = \sqrt{13^2 + (-4)^2} = \sqrt{169 + 16} = \sqrt{185}
\][/tex]
This is also correct as it matches the calculated distance [tex]\(\sqrt{185}\)[/tex].
### Conclusion:
The incorrect ways to substitute the points into the distance formula are:
- [tex]\(\sqrt{(12 - 4)^2 + (0 - (-1))^2}\)[/tex]: [tex]\(\sqrt{65}\)[/tex] differs from [tex]\(\sqrt{185}\)[/tex].
- [tex]\(\sqrt{(12 - (-1))^2 + (0 - 4)^2}\)[/tex] yields [tex]\(\sqrt{185}\)[/tex] and is correct.
Thus, the incorrect substitutions are:
- [tex]\(\sqrt{(12 - 4)^2 + (0 - (-1))^2}\)[/tex]
- \(\sqrt{(12 - -1)^2 + (0 - (-1))^2} (if it was in the question, but it wasn't)`
From the options provided, only one is incorrect (mistaken addition from point notation).
The following substitutions provided (assuming only first was incorrect) otherwise calculation holds one __Upper right __error:
[1, 2] was inferred (code).
