Question

is hljk similar to rstu? justify your answer.
yes, because a dilation by a scale factor of \\( \frac { 1 } { 3 } \\) centered at the origin maps hljk onto rstu.
yes, because a dilation by a scale factor of \\( \frac { 1 } { 3 } \\) centered at point i maps hljk onto rstu.
no, because \\( \frac { s t } { i j } \\) is not equal to \\( \frac { t u } { j k } \\).

Explanation:

Step1: Analyze the properties of dilation

Dilation is a transformation that changes the size of a figure but not its shape. For two figures to be similar, one should be a dilation of the other.

Step2: Check the center of dilation

If we consider dilation centered at point \(I\), we can observe the relationship between the corresponding sides of \(HIJK\) and \(RSTU\).

Step3: Calculate the scale factor

The distance between \(I\) and \(J\) is \(9\) units (from \(x = - 3\) to \(x=6\)), and the distance between \(I\) and \(S\) is \(3\) units (from \(x = 2\) to \(x = 6\)). The scale factor \(k=\frac{3}{9}=\frac{1}{3}\). Similarly, for other corresponding sides (e.g., vertical sides), the scale - factor calculation will also give \(\frac{1}{3}\) when centered at \(I\).

Answer:

Yes, because a dilation by a scale factor of \(\frac{1}{3}\) centered at point \(I\) maps \(HIJK\) onto \(RSTU\).