To solve for the unknown length $a$ in similar triangles, we use the property that corresponding sides of similar triangles are proportional.

Given triangles $\triangle ABC$ and $\triangle RQP$ are similar, we have the corresponding sides:

• $AB$ corresponds to $RQ$
• $AC$ corresponds to $RP$
• $BC$ corresponds to $QP$

From the problem:

• $AB = 12 \, \text{in.}$, $RQ = 30 \, \text{in.}$
• $BC = a \, \text{in.}$, $QP = b \, \text{in.}$
• $AC = 8 \, \text{in.}$, $RP = 10 \, \text{in.}$

The ratio of the corresponding sides can be written as: $\frac{AB}{RQ} = \frac{AC}{RP} = \frac{BC}{QP}$

Using the ratio of $AB$ and $RQ$: $\frac{12}{30} = \frac{8}{10}$

Now, solving for the unknown length $a$: $\frac{AB}{RQ} = \frac{BC}{QP}$ $\frac{12}{30} = \frac{a}{b}$ Since the values of $a$ and $b$ are unknown, further clarification or values for $b$ may be needed to solve for $a$.

Please confirm the value of $b$ or any additional information if available. Would you like to continue solving this with further assumptions or adjustments?

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Follow-up questions:

1. What are the specific known lengths in triangle $RQP$?
2. Can you confirm if the unknown lengths correspond to sides in the same ratio?
3. Do you have additional information on the angles of the triangles?
4. Would you like to calculate based on any other known proportion?
5. Should we make any assumptions about the unknown values?

Tip: When dealing with similar triangles, always identify the corresponding sides and set up proportional relationships between them.