- Triangle similarity \(\triangle HLI \sim \triangle JLK\) implies proportionality of all corresponding side pairs.
- By matching corresponding vertices (\(H \leftrightarrow J\), \(L \leftrightarrow L\), \(I \leftrightarrow K\)), we identify the three equal ratios of sides.
- These ratios are \(\frac{HL}{JL} = \frac{IL}{KL} = \frac{HI}{JK}\).
- Therefore, the ratio \(\frac{HL}{JL} = \frac{IL}{KL}\) is equal to \(\frac{HI}{JK}\).
- Final answer: \(\boxed{\frac{ HI }{ JK }}\)
Explanation
- Identify corresponding side ratios in similar triangles.
The definition of similarity between two triangles states that if two triangles are similar, the ratios of their corresponding sides are equal. We are given that \(\triangle HLI \sim \triangle JLK\).
By matching the vertices in the given order, we identify the corresponding sides:
- The first two vertices of \(\triangle HLI\) are \(H\) and \(L\), corresponding to vertices \(J\) and \(L\) in \(\triangle JLK\). Thus, one ratio is \(\frac{HL}{JL}\).
- The last two vertices of \(\triangle HLI\) are \(L\) and \(I\), corresponding to vertices \(L\) and \(K\) in \(\triangle JLK\). Thus, another ratio is \(\frac{LI}{LK}\), which is equal to \(\frac{IL}{KL}\).
- The first and last vertices of \(\triangle HLI\) are \(H\) and \(I\), corresponding to vertices \(J\) and \(K\) in \(\triangle JLK\). Thus, the third ratio is \(\frac{HI}{JK}\).
Since the triangles are similar, these three ratios must be equal:
\[\frac{HL}{JL} = \frac{LI}{LK} = \frac{HI}{JK}\]
Given that \(\frac{HL}{JL} = \frac{IL}{KL}\), it follows directly that this ratio is also equal to \(\frac{HI}{JK}\).
Examples
Understanding similar triangles is essential in architecture and engineering. For example, if you know the ratio of the sides of a small-scale model of a bridge, you can use those same ratios to calculate the exact dimensions required for the full-scale construction, ensuring that the structural proportions remain consistent. Similar to matching the ratios in \(\triangle HLI \sim \triangle JLK\), engineers compare these proportions to ensure safety and stability at any scale.