If the image of point P(2,3) in a line L is Q(4,5), then the image of point R(0,0) in the …

Question

If the image of point $P(2,3)$ in a line $L$ is $Q(4,5)$, then the image of point $R(0,0)$ in the same line is:

Accepted Answer

Midpoint of $PQ = (3,4)$, and $PQ$ has slope 1, so $L$ (perpendicular bisector of $PQ$) has slope $-1$ through $(3,4)$: $x+y=7$.

Image of $R(0,0)$: let it be $S(x_1,y_1)$; midpoint $\left(\dfrac{x_1}2,\dfrac{y_1}2\right)$ on $L$: $x_1+y_1=14$.

Since $RS\perp L$: slope of $RS=1 \implies y_1=x_1$.

$$2x_1=14 \implies x_1=y_1=7$$

So the image is $(7,7)$.

Solutions · 1