We are given that triangles $HKL$ and $HIJ$ are similar. We need to find which of the given ratios is equal to $\frac{LH}{JH}$.

1. Problem Description

We are given that triangles $HKL$ and $HIJ$ are similar. We need to find which of the given ratios is equal to $\frac{LH}{JH}$.

2. Solution Steps

Since $\triangle HKL \sim \triangle HIJ$, the corresponding sides are proportional.

This means:

$\frac{HK}{HI} = \frac{HL}{HJ} = \frac{KL}{IJ}$

We are looking for a ratio that is equal to $\frac{LH}{JH}$. Since $LH = HL$ and $JH = HJ$, we can rewrite the required ratio as $\frac{HL}{HJ}$.

From the similarity relation, we have $\frac{HL}{HJ} = \frac{HK}{HI}$.

Also, $\frac{HL}{HJ} = \frac{KL}{IJ}$.

Let's look at the given options:

A. $\frac{KL}{JI} = \frac{KL}{IJ}$. This is the inverse of $\frac{IJ}{KL}$. So $\frac{KL}{JI} = \frac{HL}{HJ}$. Therefore $\frac{LH}{JH} = \frac{KL}{JI}$. The options are not correct because $JI$ is the same as $IJ$, therefore A can be rewritten as $\frac{KL}{IJ}$. This is the correct match, hence $\frac{LH}{JH} = \frac{KL}{IJ}$.

B. $\frac{HK}{JK}$. This is not in the same form as $\frac{HK}{HI}$.

C. $\frac{JI}{KL}$. Since $JI$ is the same as $IJ$, this becomes $\frac{IJ}{KL}$, which is the inverse of $\frac{KL}{IJ}$. So $\frac{IJ}{KL} = \frac{HJ}{HL}$. Since $\frac{HJ}{HL}$ is equal to $\frac{1}{\frac{HL}{HJ}}$, it is the inverse of what we want. Therefore C is not the answer.

Therefore, the answer should be A. The ratio $\frac{LH}{JH}$ is equal to $\frac{KL}{JI}$ which is the same as $\frac{KL}{IJ}$.

3. Final Answer

A. $\frac{KL}{JI}$