# Scaling at an arbitrary point and figuring out the distance from origin

Suppose I have a $8 \times 6$ rectangle, with its lower left corner at the origin $\left(0, 0\right)$. I want to scale this rectangle by $\frac{1}{2}$ at an anchor point $\left(3, 3\right)$. So the resulting rectangle is $4 \times 3$, but I cannot figure out how to compute the distance from the origin to the lower left corner of the new rectangle. Help is appreciated.

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Sorry if I'm misunderstanding the construction, but doesn't that corner just go to the midpoint between the origin and (3,3)? In that case it's half the distance to (3,3), which is computed with the Pythagorean theorem. – Jonas Meyer Oct 1 '10 at 4:19
What do you mean by "anchor point"? Should the newly scaled rectangle have a corner at that point, or should it be centered there? – J. M. Oct 1 '10 at 4:20
It is the midpoint for this case, but not for all cases. By anchor point I mean the center of scaling (as if the rectangle was moved to the origin and then scaled). – Morrowless Oct 1 '10 at 4:22
Oh, I guess you're saying you really wanted a general way to find distances of transformed points. – Jonas Meyer Oct 1 '10 at 4:32

Scaling by $0.5$ about the center $(3, 3)$ is the same as translating by $(-3, -3)$, scaling by $0.5$ about $(0, 0)$, then translating by $(3, 3)$. So, to find the coordinates of the lower left corner of the new rectangle, take the coordinates of the lower left corner of the original rectangle, subtract $3$ from each coordinate, multiply each coordinate by $0.5$, and add $3$ to each coordinate. Once you have the point's coordinates, you can find its distance from the origin.