Question

select all representations of transformations that preserve distance.
(x,y)→(y,x)
(x,y)→(2x,2y)
(x,y)→(x, - y)
(x,y)→(x + 1/2,y + 1/2)
(x,y)→(1/2x,1/2y)
(x,y)→(y, - x)

Explanation:

Step1: Recall isometry concept

Transformations that preserve distance are called isometries.

Step2: Analyze option A

$(x,y)\to(y,x)$ is a reflection across $y = x$. Distance is preserved.

Step3: Analyze option B

$(x,y)\to(2x,2y)$ is a dilation with scale - factor 2. Distance is not preserved.

Step4: Analyze option C

$(x,y)\to(x, - y)$ is a reflection across the $x$ - axis. Distance is preserved.

Step5: Analyze option D

$(x,y)\to(x+\frac{1}{2},y+\frac{1}{2})$ is a translation. Distance is preserved.

Step6: Analyze option E

$(x,y)\to(\frac{1}{2}x,\frac{1}{2}y)$ is a dilation with scale - factor $\frac{1}{2}$. Distance is not preserved.

Step7: Analyze option F

$(x,y)\to(y, - x)$ is a rotation. Distance is preserved. But in the original list, this option is not correctly typed as $(x,y)\to(y,-z)$ which is incorrect. So we consider only A, C and D as correct among the given options.

Answer:

• A. $(x,y)\to(y,x)$
• C. $(x,y)\to(x, - y)$
• D. $(x,y)\to(x+\frac{1}{2},y + \frac{1}{2})$