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To solve for the missing side length \( BC \) in the two similar triangles \( \triangle ABC \) and \( \triangle PQR \), we will use the property of similar triangles: **corresponding sides are proportional**.

### Step 1: Identify the corresponding sides  
From the given triangles:
- \( AB \) corresponds to \( PQ \),
- \( AC \) corresponds to \( PR \),
- \( BC \) corresponds to \( QR \).

The lengths are:
- \( AB = 5 \), \( PQ = 20 \),
- \( AC = 6 \), \( PR = 24 \),
- \( QR = 32 \), \( BC = ? \).

### Step 2: Set up the proportion  
Using the corresponding sides \( \frac{AB}{PQ} = \frac{BC}{QR} \):
\[
\frac{5}{20} = \frac{BC}{32}
\]

### Step 3: Solve for \( BC \)  
Simplify \( \frac{5}{20} \) to \( \frac{1}{4} \):
\[
\frac{1}{4} = \frac{BC}{32}
\]

Now, solve for \( BC \) by cross-multiplying:
\[
1 \cdot 32 = 4 \cdot BC
\]
\[
BC = \frac{32}{4} = 8
\]

### Final Answer:  
The missing side \( BC \) is **8**.

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### Related Questions:
1. How do you prove two triangles are similar?
2. What are the properties of similar triangles?
3. How do you solve for an unknown side in similar triangles?
4. How do you scale triangles using a ratio?
5. Can the similarity ratio of triangles be used in real-life applications?

### Tip:
Always ensure you set up proportions carefully when solving problems involving similar triangles!

Find the missing side length in two similar triangles ΔABC and ΔPQR, given: AB = 5, AC = 6, PQ = 20, PR = 24, QR = 32, and BC = ?.

Learn how to find the missing side length BC in two similar triangles ΔABC and ΔPQR using proportionality. Given AB = 5, AC = 6, PQ = 20, PR = 24, and QR = 32, calculate BC step-by-step using triangle similarity theorems. Suitable for students in Grade 8-10.

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