Question

Line segment AB has vertices A(-3,4) and B(1,-2). A dilation, centered at the origin, is applied to AB. The image has vertices A’ (-1,4/3) and B’ (1/3,-2/3). What is the scale factor of the dilation?

1/6
1/3
3
6
(On the 2/3 fraction the negative sign is only by the 2)

Answer 1

The scale factor is
`(1)/(3)`

Explanation:

The given line segment AB has vertices A(-3,4) and B(1,-2).

The image has vertices
`A'(-1,(4)/(3))` and
`B'((1)/(3),(-2)/(3))`.

The mapping for a dilation with scale factor
`k` centered at the origin is

`(x,y)\to (kx,ky)`

Let the scale factor for the dilation be
`k`.

Then,

`A(-3,4)\to A'(-3k,4k)`

Comparing
`A'(-3k,4k)` to
`A'(-1,(4)/(3))`, we have

`-3k=-1`

`\Rightarrow k=(1)/(3)`

Or

`4k=(4)/(3)`

`\Rightarrow k=(4)/(3)* (1)/(4)`.

`\Rightarrow k=(1)/(3)`.

We could have also used the second point to obtain the same result.