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

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

Step-by-step 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) +(1)/(k)\cdot [Q'(x,y)-O(x,y)]`

`Q(x,y) =(0,0)+(1)/(2)\cdot [(16,8)-(0,0)]`

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

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