Question

select all representations of transformations that preserve distance.
(x,y)→(y,x)
(x,y)→(2x,2y)
(x,y)→(\frac{1}{2}x,\frac{1}{2}y)
(x,y)→(x, - y)
(x,y)→(y, - x)
(x,y)→(x + \frac{1}{2},y + \frac{1}{2})

Explanation:

Step1: Recall distance - preserving transformations

Distance - preserving transformations (isometries) include reflections, rotations, and translations.

Step2: Analyze each transformation

1. For $(x,y)\to(y,x)$, it is a reflection over the line $y = x$. It preserves distance.
2. For $(x,y)\to(2x,2y)$, it is a dilation with a scale factor of 2. It does not preserve distance.
3. For $(x,y)\to(\frac{1}{2}x,\frac{1}{2}y)$, it is a dilation with a scale factor of $\frac{1}{2}$. It does not preserve distance.
4. For $(x,y)\to(x, - y)$, it is a reflection over the $x$ - axis. It preserves distance.
5. For $(x,y)\to(y,-x)$, it is a rotation of $270^{\circ}$ counter - clockwise about the origin. It preserves distance.
6. For $(x,y)\to(x+\frac{1}{2},y+\frac{1}{2})$, it is a translation. It preserves distance.

Answer:

$(x,y)\to(y,x)$
$(x,y)\to(x, - y)$
$(x,y)\to(y,-x)$
$(x,y)\to(x+\frac{1}{2},y+\frac{1}{2})$