Question

which of the following is not a true statement about the dilation shown? the ratio of \\( \frac { h ^ { prime } i ^ { prime } } { h i } = \frac { i ^ { prime } j ^ { prime } } { i j } \\) the dilation changed the orientation of the figure. the dilation represents an enlargement. the dilation can be represented by \\( ( x, y ) \
ightarrow ( 2 x, 2 y ) \\)

Explanation:

Step1: Analyze the ratio

For dilation, corresponding sides are proportional. If \(HI\) and \(IJ\) are sides of the original figure and \(H'I'\) and \(I'J'\) are sides of the dilated figure, since dilation is a similarity transformation, \(\frac{H'I'}{HI}=\frac{I'J'}{IJ}\) (because in dilation, the scale factor is the same for all corresponding sides).

Step2: Check orientation

Dilation is a similarity transformation that does not change the orientation of the figure. The shape of the figure (in terms of the order of its vertices and the angles between its sides) remains the same.

Step3: Check if it is an enlargement

If the scale factor \(k> 1\), it is an enlargement. If we assume a point \((x,y)\) in the original figure and \((2x,2y)\) in the dilated figure (from the transformation \((x,y)\to(2x,2y)\)), the scale factor \(k = 2>1\), so it is an enlargement.

Step4: Check the transformation rule

If we take a vertex, say \(H(-2,2)\) in the original figure and \(H'(-4,4)\) in the dilated figure. Substituting into \((x,y)\to(2x,2y)\): when \(x=-2,y = 2\), \(2x=-4\) and \(2y = 4\). Similarly, for \(I(2,2)\) to \(I'(4,4)\), \(J(2,-2)\) to \(J'(4,-4)\) and \(K(-2,-2)\) to \(K'(-4,-4)\) the transformation \((x,y)\to(2x,2y)\) holds.

Answer:

The dilation changed the orientation of the figure.