Question

figure ( a b ^ { prime } c ^ { prime } ) is a dilation of figure ( a b c ). vertex ( a ) is the center of dilation. complete the statement. write your answer in the blank.
figure ( a b c sim ) figure
emily draws figures ( p , q , r ), and ( s ). the center of dilation is point ( z ). which figures are similar to figure ( q ) ? explain your reasoning.
mrs. sloan draws two figures on the board. allen says that figure ( z ) is dilated to form figure ( z ^ { prime } ) using a scale factor of 3 and point ( x ) as the center of dilation. what mistake did allen make? what is the correct relationship? explain your reasoning.

Explanation:

1.

Step1: Recall the property of dilation

Dilation creates similar figures. If \(AB'C'\) is a dilation of \(ABC\) with center \(A\), then the two - triangle figures are similar. The notation for similar figures is \(\sim\). Since \(AB'C'\) is the dilated figure of \(ABC\), by the definition of similar figures formed by dilation, we have the relationship.

Figures \(P\), \(R\), and \(S\) are similar to figure \(Q\). Because dilation is a transformation that changes the size of a figure but not its shape. All the figures \(P\), \(Q\), \(R\), and \(S\) are squares (same shape), and since they are formed by dilation (with center \(Z\)), they are similar.

Allen's mistake: If the scale factor is \(3\), the figure should be larger. But \(Z'\) is smaller than \(Z\). The correct relationship: Figure \(Z'\) is a dilation of figure \(Z\) with a scale factor less than \(1\) (by counting the side - length ratios on the grid, assume side - length of \(Z\) is \(6\) units and of \(Z'\) is \(2\) units, scale factor \(k=\frac{2}{6}=\frac{1}{3}\)). Because dilation with \(0 < k<1\) makes the figure smaller.

Answer:

\(AB'C'\)

2.