Question

abcde ~ ptsrq. find pq and bc.
write your answers as decimals or whole numbers.
pq =
bc =

Explanation:

Step1: Determine the scale factor

Since the pentagons \( ABCDE\sim PTSRQ\), the ratio of corresponding sides is equal. Let's find the scale factor using the sides \( DE = 21\) and \( ST=27\). The scale factor \(k\) of \(ABCDE\) to \(PTSRQ\) is \(\frac{DE}{ST}=\frac{21}{27}=\frac{7}{9}\). The scale factor of \(PTSRQ\) to \(ABCDE\) is \(\frac{9}{7}\).

Step2: Find \(PQ\)

The side \(EA = 14\) in pentagon \(ABCDE\) corresponds to \(RQ = 27\) in pentagon \(PTSRQ\). Let \(PQ\) correspond to \(AB\) (not labeled with a number, but using the fact that for similar polygons, if we consider the ratio of \(EA\) to \(RQ\) and assume the correspondence. Wait, no, actually, since \(EA\) and \(RQ\):
We know that if we use the scale factor. Let's use the correct correspondence. \(EA\) in \(ABCDE\) and \(RQ\) in \(PTSRQ\). The length of \(EA = 14\) and \(RQ = 27\). But wait, no, actually, the scale - factor.
Since \(ABCDE\sim PTSRQ\), the ratio of corresponding sides is the same. Let's assume \(EA\) corresponds to \(RQ\). But actually, the correct correspondence: \(DE\) corresponds to \(ST\), \(EA\) corresponds to \(RQ\), \(AB\) corresponds to \(QP\), \(BC\) corresponds to \(PR\), \(CD\) corresponds to \(TS\).
We know that \(\frac{EA}{RQ}=\frac{DE}{ST}\). To find \(PQ\):
Since \(EA = 14\) and \(RQ = 27\) (incorrect, wait no, \(EA = 14\), and if we use the scale factor. Wait, no, actually, the side \(EA\) in \(ABCDE\) and \(RQ\) in \(PTSRQ\) (no, wrong correspondence). The correct correspondence: \(EA\) in \(ABCDE\) and \(RQ\) in \(PTSRQ\) (no). Wait, the similarity statement \(ABCDE\sim PTSRQ\) means \(A\) corresponds to \(P\), \(B\) corresponds to \(T\), \(C\) corresponds to \(S\), \(D\) corresponds to \(R\), \(E\) corresponds to \(Q\). So \(EA\) corresponds to \(QP\).
We know that \(\frac{EA}{QP}=\frac{DE}{RQ}\). Given \(EA = 14\), \(DE = 21\), \(RQ = 27\).

$$ \frac{14}{PQ}=\frac{21}{27} $$

Cross - multiply: \(21\times PQ=14\times27\). Then \(PQ=\frac{14\times27}{21}=\frac{378}{21} = 18\).

Step3: Find \(BC\)

Since \(BC\) corresponds to \(PR\) (and \(PR = 27\)). Using the scale factor \(\frac{DE}{ST}=\frac{21}{27}=\frac{7}{9}\) (scale factor of \(ABCDE\) to \(PTSRQ\)). So \(\frac{BC}{PR}=\frac{7}{9}\). Since \(PR = 27\), then \(BC=\frac{7}{9}\times27=21\).

Answer:

\(PQ = 18\), \(BC = 21\)