Answer :
Given that [tex]\(\triangle HLI \sim \triangle JLK\)[/tex] by the SSS (Side-Side-Side) similarity theorem, we know that the corresponding sides of similar triangles are proportional.
In this situation, let's denote the sides of [tex]\(\triangle HLI\)[/tex] and [tex]\(\triangle JLK\)[/tex] as follows:
- [tex]\(\triangle HLI\)[/tex] has sides [tex]\(H-L\)[/tex], [tex]\(L-I\)[/tex], and [tex]\(H-I\)[/tex].
- [tex]\(\triangle JLK\)[/tex] has sides [tex]\(J-L\)[/tex], [tex]\(L-K\)[/tex], and [tex]\(J-K\)[/tex].
Since the triangles are similar by the SSS similarity theorem, the ratios of their corresponding sides are equal. Therefore, we can write the following ratio relationships:
[tex]\[ \frac{HL}{JL} = \frac{IL}{KL} = \frac{HI}{JK} \][/tex]
Now let's check the given options to determine which one matches the equal ratio [tex]\( \frac{HL}{JL} = \frac{IL}{KL} \)[/tex]:
- [tex]\(\frac{HI}{JK}\)[/tex]
Here, [tex]\( \frac{HI}{JK} \)[/tex] matches the ratio of corresponding sides.
Therefore, the ratio that is equal to [tex]\(\frac{HL}{JL} = \frac{IL}{KL}\)[/tex] is:
[tex]\[ \frac{HI}{JK} \][/tex]
Thus, the correct answer is that [tex]\(\frac{HL}{JL} = \frac{IL}{KL}\)[/tex] is also equal to [tex]\(\frac{HI}{JK}\)[/tex].
[tex]\[ \boxed{3} \][/tex]
In this situation, let's denote the sides of [tex]\(\triangle HLI\)[/tex] and [tex]\(\triangle JLK\)[/tex] as follows:
- [tex]\(\triangle HLI\)[/tex] has sides [tex]\(H-L\)[/tex], [tex]\(L-I\)[/tex], and [tex]\(H-I\)[/tex].
- [tex]\(\triangle JLK\)[/tex] has sides [tex]\(J-L\)[/tex], [tex]\(L-K\)[/tex], and [tex]\(J-K\)[/tex].
Since the triangles are similar by the SSS similarity theorem, the ratios of their corresponding sides are equal. Therefore, we can write the following ratio relationships:
[tex]\[ \frac{HL}{JL} = \frac{IL}{KL} = \frac{HI}{JK} \][/tex]
Now let's check the given options to determine which one matches the equal ratio [tex]\( \frac{HL}{JL} = \frac{IL}{KL} \)[/tex]:
- [tex]\(\frac{HI}{JK}\)[/tex]
Here, [tex]\( \frac{HI}{JK} \)[/tex] matches the ratio of corresponding sides.
Therefore, the ratio that is equal to [tex]\(\frac{HL}{JL} = \frac{IL}{KL}\)[/tex] is:
[tex]\[ \frac{HI}{JK} \][/tex]
Thus, the correct answer is that [tex]\(\frac{HL}{JL} = \frac{IL}{KL}\)[/tex] is also equal to [tex]\(\frac{HI}{JK}\)[/tex].
[tex]\[ \boxed{3} \][/tex]
