To solve this, we'll need to use the property of similar triangles. For two triangles that are similar, corresponding sides are proportional. In this case, we have two triangles ABC and RQP, where side lengths of both triangles are given and labeled.

Let's denote the corresponding sides of the triangles:

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

Given information:

• $AB = 12 \, \text{in}$,
• $BC = 30 \, \text{in}$,
• $ab = 8 \, \text{in}$ (corresponding to $AB$),
• $ac = 10 \, \text{in}$.

Since the triangles are similar, the sides will have the following proportion:

$\frac{AB}{ab} = \frac{AC}{ac} = \frac{BC}{bc}$

Step 1: Find the ratio of the sides

Using the sides $AB$ and $ab$: $\frac{AB}{ab} = \frac{12}{8} = \frac{3}{2}$

Step 2: Use the ratio to find other sides

Now, let's find the unknown side $bc$ (the corresponding side of $BC$):

$\frac{BC}{bc} = \frac{3}{2}$ $\frac{30}{bc} = \frac{3}{2}$

Cross-multiply to solve for $bc$:

$30 \times 2 = 3 \times bc$ $60 = 3 \times bc$ $bc = \frac{60}{3} = 20 \, \text{in}$

Final Answer:

• The unknown length $bc$ is 20 inches.

Let me know if you need any further details or explanations!

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5 Related Questions:

1. What is the definition of similar triangles?
2. How do you determine the corresponding sides of similar triangles?
3. Can you apply this method if the triangles are not labeled?
4. How do you prove two triangles are similar?
5. What are the other types of triangle similarity rules (e.g., AA, SSS)?

Tip:

When dealing with similar triangles, always match corresponding sides before setting up your proportion.