Question

If the vertex of image Q’ is located at the coordinates (16,8) was dilated by the scale factor of 2, what was the coordinates for the pre-image Q?

Answer 1

Answer:

The pre-image Q is $Q(x,y) =(8,4)$ .

Explanation:

From Linear Algebra, we can define dilation by the following expression:

$Q'(x,y) = O(x,y) + k\cdot [Q(x,y)-O(x,y)]$ (1)

Where:

$O(x,y)$ - Reference point, dimensionless.

$Q(x,y)$ - Original point, dimensionless.

$Q'(x,y)$ - Dilated point, dimensionless.

$k$ - Dilation factor, dimensionless.

If we know that $O(x,y) = (0,0)$, $Q'(x,y) = (16,8)$ and $k = 2$, then the original point is:

$Q'(x,y)-O(x,y) = k\cdot [Q(x,y)-O(x,y)]$

$Q(x,y) = O(x,y) +\frac{1}{k}\cdot [Q'(x,y)-O(x,y)]$

$Q(x,y) =(0,0)+\frac{1}{2}\cdot [(16,8)-(0,0)]$

$Q(x,y) =(8,4)$

The pre-image Q is $Q(x,y) =(8,4)$ .