ChatGPT's answer to 2D geometry

hbghlyj

2D geometry
Calculate reflection of the coordinate points (z,z-0.1), (z,z), (1,z) over the line x+y=1
Only give result; don't show steps

The output of ChatGPT:

The reflections of the points are:

    (z-0.05, z+0.05)
    (z, z+0.1)
    (z+0.1, 0.9-z)

None of its answers is correct

Let's check using Mathematica:

1. In[]:= ReflectionTransform[{1,1},{1,0}]/@{{z,z-0.1},{z,z},{1,z}}
2. Out[]= {{1.1 -z,1-z},{1-z,1-z},{1-z,0}}

复制代码

Documentation for ReflectionTransform:
    ReflectionTransform[v,p] gives a reflection in a mirror through the point $p$, normal to the vector $v$

hbghlyj

Using the formula $(x, y)⟶(1 - y, 1 - x)$ your second answer is wrong

ChatGPT: I apologize for the mistake in my previous response. You are correct that the formula to reflect a point over the line x+y=1 is $(x,y) \rightarrow (1-y, 1-x)$, and I made an error in my calculation.
Here are the corrected reflections of the points:

    $(z-0.05, z+0.05)$
    $(z+0.1, z)$
    $(0.9-z, 1-z)$

I apologize for any confusion that my previous response may have caused. Thank you for bringing it to my attention.

Plugging $x=z,y=z+0.1$ into $(1-y,1-x)$, what do you get? your second answer is still wrong

ChatGPT: I apologize again for the mistake in my previous response. You are correct that I made another error in my calculation.

Using the formula $(x,y) \rightarrow (1-y, 1-x)$ to reflect the points over the line $x+y=1$, and plugging in the coordinates $(z,z-0.1)$, $(z,z)$, and $(1,z)$, we get:

    $(z+0.1, z)$
    $(z, z)$
    $(0.9-z, 1-z)$

Again, I apologize for any confusion caused by my previous responses. Thank you for bringing these errors to my attention.

still wrong