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

Answer:

The scale factor is $\frac{1}{3}$

Step-by-step explanation:

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

The image has vertices $A'(-1,\frac{4}{3})$ and $B'(\frac{1}{3},\frac{-2}{3})$.

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  $A'(-1,\frac{4}{3})$, we have

$-3k=-1$

$\Rightarrow k=\frac{1}{3}$

Or

$4k=\frac{4}{3}$

$\Rightarrow k=\frac{4}{3}\times \frac{1}{4}$.

$\Rightarrow k=\frac{1}{3}$.

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