Find the unknown lengths in the pair of similar triangles.
ABCRQP12 in.30 in.ab8 in.10 in.
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Part 1
The length of a
 
is enter your response here
 
in. 

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!

---
### 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.

Find the unknown lengths in the pair of similar triangles. AB = 12 in., BC = 30 in., ab = 8 in., ac = 10 in. What is the length of bc?

Learn how to find the unknown side lengths in similar triangles using proportions. This problem covers key geometric concepts, including similar triangles, proportional sides, and the Triangle Similarity Theorem. Suitable for students in Grades 8-10.

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