Question

Dilate the the triangle, scale factor = 3 (positive 3, ignore the

negative in the image below) with a center of dilation about
the origin.
A(-7,-6)
B(-5,-2)
C(-1,-5)

Answer 1

The rule of dilation is
`P'(x,y) = 3\cdot P(x,y)`.

The vertices of the dilated triangle are
`A'(x,y) = (-21, -18)`,
`B'(x,y) = (-15,-10)` and
`C'(x,y) = (-3,-15)`, respectively.

Explanation:

From Linear Algebra, we define the dilation by the following definition:

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

Where:

`O(x,y)` - Center of dilation, dimensionless.

`k` - Scale factor, dimensionless.

`P(x,y)` - Original point, dimensionless.

`P'(x,y)` - Dilated point, dimensionless.

If we know that
`O(x,y) = (0,0)`,
`k = 3`,
`A(x,y) = (-7,-6)`,
`B(x,y) = (-5,-2)` and
`C(x,y) =(-1,-5)`, then dilated points of triangle ABC are, respectively:

`A'(x,y) = O(x,y) + k\cdot [A(x,y)-O(x,y)]` (2)

`A'(x,y) = (0,0) + 3\cdot [(-7,-6)-(0,0)]`

`A'(x,y) = (-21, -18)`

`B'(x,y) = O(x,y) + k\cdot [B(x,y)-O(x,y)]` (3)

`B'(x,y) = (0,0) + 3\cdot [(-5,-2)-(0,0)]`

`B'(x,y) = (-15,-10)`

`C'(x,y) = O(x,y) + k\cdot [C(x,y)-O(x,y)]` (4)

`C'(x,y) = (0,0) +3\cdot [(-1,-5)-(0,0)]`

`C'(x,y) = (-3,-15)`

The rule of dilation is:

`P'(x,y) = 3\cdot P(x,y)` (5)

The vertices of the dilated triangle are
`A'(x,y) = (-21, -18)`,
`B'(x,y) = (-15,-10)` and
`C'(x,y) = (-3,-15)`, respectively.