Question

a pool company is creating a blueprint for a family pool and a similar dog pool for a new client. which statement explains how the company can determine whether pool lmno is similar to pool pqrs?
translate pqrs so that point p of pqrs lies on point l of lmno, then dilate pqrs by the ratio (\frac{lm}{pq})
translate pqrs so that point q of pqrs lies on point m of lmno, then dilate pqrs by the ratio (\frac{pq}{lm})
translate pqrs so that point p of pqrs lies on point l of lmno. then translate pqrs so that point q of pqrs lies on point m of lmno
translate pqrs so that point q of pqrs lies on point m of lmno. then translate pqrs so that point p of pqrs lies on point l of lmno

Explanation:

Step1: Recall similarity transformation

Similarity of polygons can be checked by translation (to align a corresponding vertex) and then dilation (by the ratio of corresponding side lengths).
For polygons \(LMNO\) and \(PQRS\), we first translate \(PQRS\) so that a corresponding vertex (say \(P\) which corresponds to \(L\)) is aligned. Then we check the ratio of side lengths. The ratio of side lengths of similar polygons is constant. If we translate \(PQRS\) so that \(P\) (a vertex of \(PQRS\)) lies on \(L\) (a vertex of \(LMNO\)), and then dilate \(PQRS\) by the ratio \(\frac{LM}{PQ}\) (where \(LM\) is a side of \(LMNO\) and \(PQ\) is a side of \(PQRS\)). If after dilation all corresponding vertices and sides match (in terms of shape and proportion), the polygons are similar.

Step2: Analyze other options

• Translating \(PQRS\) so that \(Q\) lies on \(M\) first (without aligning the starting - point vertex in the correct correspondence) is not the standard way of checking similarity. Similarity transformations start with aligning a pair of corresponding vertices.
• Just translating \(PQRS\) multiple times without a dilation (which is necessary for size - proportion adjustment in similarity) (options with only translations) will not check for similarity as similarity requires both congruence - like (after translation) and size - proportion (after dilation) conditions.

Answer:

Translate \(PQRS\) so that point \(P\) of \(PQRS\) lies on point \(L\) of \(LMNO\), then dilate \(PQRS\) by the ratio \(\frac{LM}{PQ}\)