Question

the quadrilaterals abcd and jklm are similar. find the length x of \\(\overline{kl}\\).

Explanation:

Step1: Determine the scale factor

Since the quadrilaterals are similar, the ratio of corresponding sides is equal. First, find the scale factor using the sides \(AB = 3\) and \(JK = 4.5\). The scale factor \(k=\frac{JK}{AB}=\frac{4.5}{3}=1.5\).

Step2: Calculate the length of \(KL\)

The corresponding side to \(KL\) in \(ABCD\) is \(BC = 7\). To find \(KL\), multiply the length of \(BC\) by the scale factor. So \(x = BC\times k=7\times1.5 = 10.5\)? Wait, no, wait. Wait, let's check the other corresponding sides. Wait, maybe I mixed up the corresponding sides. Let's check the sides \(AD = 3\) and \(JM = 4.5\)? Wait, no, in the first quadrilateral, \(AD = 3\), \(JM = 4.5\)? Wait, no, the first quadrilateral: \(AB = 3\), \(BC = 7\), \(CD = 2\), \(DA = 3\). The second quadrilateral: \(JK = 4.5\), \(KL = x\), \(LM = 1.8\), \(MJ = 4.5\)? Wait, no, \(JM = 4.5\)? Wait, \(CD = 2\) and \(LM = 1.8\)? Wait, no, that can't be. Wait, maybe the corresponding sides are \(CD = 2\) and \(LM = 1.8\)? Wait, no, let's re - examine.

Wait, maybe the correct corresponding sides: Let's see, in quadrilateral \(ABCD\) and \(JKLM\), the order of the vertices matters. So \(AB\) corresponds to \(JK\), \(BC\) corresponds to \(KL\), \(CD\) corresponds to \(LM\), and \(DA\) corresponds to \(MJ\).

So \(CD = 2\), \(LM = 1.8\). Wait, no, \(CD = 2\), \(LM = 1.8\)? Then the scale factor would be \(\frac{LM}{CD}=\frac{1.8}{2}=0.9\), but \(JK = 4.5\), \(AB = 3\), \(\frac{4.5}{3}=1.5\), which is a contradiction. Wait, I must have misidentified the corresponding sides.

Wait, let's look at the lengths: \(AB = 3\), \(JK = 4.5\); \(AD = 3\), \(JM = 4.5\); \(CD = 2\), \(LM = 3\)? Wait, no, \(LM = 1.8\). Wait, maybe \(CD = 2\) and \(LM = 1.8\) is wrong. Wait, the first quadrilateral: \(CD = 2\), the second quadrilateral: \(LM = 1.8\). Wait, no, maybe the correct corresponding sides are \(AB\) and \(JK\), \(BC\) and \(KL\), \(CD\) and \(LM\), \(DA\) and \(MJ\).

So \(AB = 3\), \(JK = 4.5\) (ratio \(4.5/3 = 1.5\)); \(CD = 2\), \(LM = 1.8\) (ratio \(1.8/2=0.9\)) - that's inconsistent. Wait, maybe I made a mistake in the figure. Wait, maybe \(CD = 2\) and \(LM = 3\)? No, the figure shows \(LM = 1.8\). Wait, maybe the sides \(AD = 3\) and \(JM = 4.5\) (ratio \(4.5/3 = 1.5\)), \(CD = 2\) and \(LM = 3\) (ratio \(3/2 = 1.5\)), \(AB = 3\) and \(JK = 4.5\) (ratio \(1.5\)), then \(BC = 7\) and \(KL = x\) (ratio \(1.5\)). Then \(x=7\times1.5 = 10.5\)? But \(LM\) should be \(2\times1.5 = 3\), but the figure shows \(LM = 1.8\). Wait, maybe the figure has a typo, or I misread the lengths. Wait, maybe \(CD = 2\) and \(LM = 3\) is wrong, and \(CD = 2\) and \(LM = 1.8\) is correct. Then the scale factor is \(1.8/2 = 0.9\), then \(JK\) should be \(3\times0.9=2.7\), but it's \(4.5\). So there is a mistake. Wait, maybe the sides \(AB = 3\), \(JK = 4.5\); \(BC = 7\), \(KL = x\); \(CD = 2\), \(LM = 3\) (since \(2\times1.5 = 3\), but the figure shows \(1.8\), maybe it's a mistake in the figure). Alternatively, maybe the corresponding sides are \(AB\) and \(JK\), \(BC\) and \(KL\), \(CD\) and \(LM\), \(DA\) and \(MJ\). So \(AB = 3\), \(JK = 4.5\) (scale factor \(4.5/3 = 1.5\)), \(CD = 2\), \(LM = 3\) (since \(2\times1.5 = 3\), but the figure shows \(1.8\), maybe it's a misprint). Then \(BC = 7\), so \(KL=7\times1.5 = 10.5\). Wait, but let's check with \(CD\) and \(LM\). If \(CD = 2\), \(LM = 3\), then \(3/2 = 1.5\), which matches the scale factor from \(AB\) and \(JK\). So maybe the \(LM\) in the figure is misprinted, and it should be \(3\) instead of \(1.8\). So proceeding with scale factor \(1.5\).

So \(x = 7\times1.5=10.5\)? Wait, no, wait, maybe I got the corresponding sides wrong. Let's try another approach. Let's find the ratio of \(JK\) to \(AB\): \(JK = 4.5\), \(AB = 3\), so ratio \(r=\frac{4.5}{3}=1.5\). Now, the side corresponding to \(KL\) is \(BC = 7\), so \(KL=BC\times r = 7\times1.5 = 10.5\). Alternatively, if we use the sides \(CD = 2\) and \(LM = 1.8\), the ratio is \(1.8/2 = 0.9\), then \(JK\) should be \(3\times0.9 = 2.7\), which is not \(4.5\). So the correct ratio is \(1.5\), so \(x = 10.5\). Wait, but maybe the corresponding side to \(KL\) is not \(BC\). Wait, maybe \(BC\) corresponds to \(KL\), \(AB\) corresponds to \(JK\), \(AD\) corresponds to \(JM\), \(CD\) corresponds to \(LM\). So \(AB = 3\), \(JK = 4.5\) (ratio \(1.5\)), \(AD = 3\), \(JM = 4.5\) (ratio \(1.5\)), \(CD = 2\), \(LM = 3\) (ratio \(1.5\)), \(BC = 7\), \(KL = 10.5\) (ratio \(1.5\)). So the answer should be \(10.5\). Wait, but let's check again.

Wait, maybe the figure has \(CD = 2\) and \(LM = 1.8\), but that would mean the scale factor is \(1.8/2=0.9\), and \(JK = 3\times0.9 = 2.7\), which is not \(4.5\). So there is a contradiction. So the only way the scale factor is consistent is if \(LM = 3\) (since \(2\times1.5 = 3\)) and the \(1.8\) is a misprint. So we proceed with scale factor \(1.5\).

Step1: Find the scale factor

The ratio of corresponding sides \(AB\) (length \(3\)) and \(JK\) (length \(4.5\)) is \(\frac{JK}{AB}=\frac{4.5}{3} = 1.5\).

Step2: Calculate \(KL\)

The side \(BC\) in \(ABCD\) has length \(7\) and corresponds to \(KL\) in \(JKLM\). Using the scale factor, \(KL=BC\times1.5=7\times1.5 = 10.5\).

Answer:

\(x = 10.5\)