Let L be the line y=2x, in the two dimensional plane.

Question

Let $L$ be the line $y=2x$, in the two dimensional plane.

Statement 1: The image of the point $(0,1)$ in $L$ is the point $\left(\dfrac45,\dfrac35\right)$

Statement 2: The points $(0,1)$ and $\left(\dfrac45,\dfrac35\right)$ lie on opposite sides of the line $L$ and are at equal distance from it.

Accepted Answer

Reflecting $(0,1)$ in $2x-y=0$:
$$\frac{x_1-0}2=\frac{y_1-1}{-1}=\frac{-2(2(0)-1)}{5}=\frac25$$
$$x_1=\frac45, \quad y_1=1-\frac25=\frac35$$

So Statement 1 is true. Since reflection across a line always places the image and original on opposite sides at equal perpendicular distance, Statement 2 is true and is exactly why Statement 1 holds.

Solutions · 1