Question

the quotients of the corresponding side lengths in figures jklm and pqrs are shown below the figures. (figures: jk=30 mm, kl=21 mm, lm=10 mm, mj=24 mm; pq=24 mm, qr=15 mm, rs=8 mm, sp=18 mm. ratios: ( \frac{jk}{pq} = \frac{30}{24} ), ( \frac{kl}{qr} = \frac{21}{15} ), ( \frac{lm}{rs} = \frac{10}{8} ), ( \frac{mj}{sp} = \frac{24}{18} )) is figure jklm a dilation of figure pqrs? choose one option from each drop - down menu to answer the question. in figures jklm and pqrs, the quotient of sides jk and pq is equivalent to all of the other quotients. therefore, the quotients of the corresponding side lengths of the figures show that figure jklm is not a dilation of figure pqrs because all of the quotients are equivalent.

Explanation:

Step1: Simplify \(\frac{JK}{PQ}\)

\(JK = 30\) mm, \(PQ = 24\) mm, so \(\frac{JK}{PQ}=\frac{30}{24}=\frac{5}{4} = 1.25\)

Step2: Simplify \(\frac{KL}{QR}\)

\(KL = 21\) mm, \(QR = 15\) mm, so \(\frac{KL}{QR}=\frac{21}{15}=\frac{7}{5}=1.4\)

Step3: Simplify \(\frac{LM}{RS}\)

\(LM = 10\) mm, \(RS = 8\) mm, so \(\frac{LM}{RS}=\frac{10}{8}=\frac{5}{4}=1.25\)

Step4: Simplify \(\frac{MJ}{SP}\)

\(MJ = 24\) mm, \(SP = 18\) mm, so \(\frac{MJ}{SP}=\frac{24}{18}=\frac{4}{3}\approx1.33\)

We can see that the quotients \(\frac{JK}{PQ}\), \(\frac{KL}{QR}\), \(\frac{LM}{RS}\), \(\frac{MJ}{SP}\) are not all equal. For a dilation, the ratio of corresponding sides (the scale factor) must be constant. Since the quotients of corresponding side lengths are not all equal, figure \(JKLM\) is not a dilation of figure \(PQRS\).

Answer:

• The quotient of sides \(JK\) and \(PQ\) is not equivalent to all of the other quotients.
• Therefore, the quotients of the corresponding side lengths of the figures show that figure \(JKLM\) is not a dilation of figure \(PQRS\) because not all of the quotients are equivalent.