Question

two w - shaped figures with labeled points: left figure has points a, b, c, d, e with segment ab labeled 6. right figure has points h, i, j, k, l with segment hi labeled 15.

Explanation:

Assuming the figures are similar (since they have the same "W" shape), we can use the concept of similar figures (corresponding sides are proportional). Let's denote the length of \( LK \) (corresponding to \( AB = 6 \) and \( HI = 15 \)) as \( x \).

Step 1: Identify corresponding sides

The segments \( AB \) and \( HI \) are corresponding, \( AB = 6 \), \( HI = 15 \). We need to find the length of \( LK \) (corresponding to \( AB \) or \( HI \)'s similar segment). Since the figures are similar, the ratio of corresponding sides is equal. Let's assume the ratio of similarity is \( \frac{HI}{AB}=\frac{15}{6}=\frac{5}{2} \).

Step 2: Apply the ratio to find \( LK \)

If \( AB = 6 \) and the ratio is \( \frac{5}{2} \), then \( LK = AB\times\frac{5}{2}=6\times\frac{5}{2}=15 \)? Wait, no, maybe \( AB \) and \( LK \) are corresponding? Wait, maybe the first figure has \( AB = 6 \), and the second has \( HI = 15 \), so the scale factor is \( \frac{15}{6}=\frac{5}{2} \). If we assume the sides \( AB \) and \( LK \) are corresponding, or maybe \( AB \) and \( HI \) are corresponding, and we need to find the length of \( LK \) (corresponding to \( AB \)). Wait, maybe the first figure: \( AB = 6 \), and the second figure: \( HI = 15 \), so the scale factor is \( \frac{15}{6}=\frac{5}{2} \). So if we consider the sides, the length of \( LK \) (corresponding to \( AB \)) would be \( 6\times\frac{5}{2}=15 \)? Wait, no, maybe I got the correspondence wrong. Alternatively, if the first figure's side \( AB = 6 \) and the second's \( HI = 15 \), then the scale factor is \( \frac{15}{6}=\frac{5}{2} \). So if we need to find the length of \( LK \), which is corresponding to \( AB \), then \( LK = 6\times\frac{5}{2}=15 \). Wait, but maybe the question is to find the length of \( LK \) given that the figures are similar. Let's recheck.

Wait, the first figure: points \( A, B, C, D, E \) with \( AB = 6 \). The second figure: \( H, I, J, K, L \) with \( HI = 15 \). Since the figures are similar, the ratio of \( HI \) to \( AB \) is \( \frac{15}{6}=\frac{5}{2} \). So the corresponding side \( LK \) (to \( AB \)) would be \( AB\times\frac{5}{2}=6\times\frac{5}{2}=15 \)? Wait, no, maybe \( AB \) and \( LK \) are corresponding, so \( LK = 15 \). Alternatively, maybe the length of \( LK \) is 15. Wait, maybe the problem is to find the length of \( LK \) when the two figures are similar, with \( AB = 6 \) and \( HI = 15 \). So using similar triangles (or similar polygons) ratio: \( \frac{HI}{AB}=\frac{LK}{AB} \)? No, wait, \( AB \) and \( HI \) are corresponding, \( BD \) and \( IK \) (but maybe not). Wait, perhaps the key is that the two "W" shapes are similar, so the ratio of corresponding sides is equal. So \( \frac{HI}{AB}=\frac{LK}{AB} \)? No, that doesn't make sense. Wait, \( AB = 6 \), \( HI = 15 \), so the scale factor is \( \frac{15}{6}=\frac{5}{2} \). So if \( AB \) is 6, then \( LK \) (corresponding to \( AB \)) is \( 6\times\frac{5}{2}=15 \). So the length of \( LK \) is 15.

Answer:

15