Question

R(3, 8) is dilated by a factor of 5, centered at (2, 10). Write the coordinates of R’.

Answer 1

The coordinates of R' are
`R'(x,y) = (8,-2)`.

Explanation:

According to Linear Algebra, dilation consist in expanding a given vector with respect to one of its endpoints by a scalar. That is:

`R'(x, y) = C(x,y) + k\cdot [R(x,y)-C(x,y)]` (Eq. 1)

Where:

`R(x, y)` - Original location with respect to origin, dimensionless.

`C(x,y)` - Point of reference with respect to origin, dimensionless.

`k` - Dilation factor, dimensionless.

`R'(x, y)` - Dilated point with respect to origin, dimensionless.

If we know that
`R(x,y) = (3,8)`,
`C(x,y) = (2,10)` and
`k = 5`, then the dilated point is:

`R'(x,y) = (3,8) +5\cdot [(3,8)-(2,10)]`

`R'(x,y) = (3,8)+5\cdot (1,-2)`

`R'(x,y) = (8,-2)`

The coordinates of R' are
`R'(x,y) = (8,-2)`.