If [tex]\triangle HLI \sim \triangle JLK[/tex] by the SSS similarity theorem, then [tex]\frac{HL}{JL} = \frac{IL}{KL}[/tex] is also equal to which ratio?

A. [tex]\frac{HI}{JK}[/tex]
B. [tex]\frac{HJ}{J}[/tex]
C. [tex]\frac{IK}{KL}[/tex]
D. [tex]\frac{IK}{HI}[/tex]



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]