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The image contains a mathematical problem written in Gujarati. The problem involves two triangles, \( \triangle ABC \) and \( \triangle PQR \), and discusses their corresponding sides being in proportion. The given information is as follows:

\[ \frac{AB}{PQ} = \frac{BC}{QR} = \frac{AC}{PR} \]

The lengths of the sides are given:
- \( AB = 15 \) cm
- \( PQ = 20 \) cm
- \( BC = 12 \) cm

The problem asks to find the length of \( QR \).

### Solution:
Given the proportion:

\[ \frac{AB}{PQ} = \frac{BC}{QR} \]

Plugging in the known values:

\[ \frac{15}{20} = \frac{12}{QR} \]

We need to solve for \( QR \). First, simplify the ratio on the left side:

\[ \frac{15}{20} = \frac{3}{4} \]

Now, set up the equation:

\[ \frac{3}{4} = \frac{12}{QR} \]

To solve for \( QR \), cross-multiply:

\[ 3 \cdot QR = 4 \cdot 12 \]

\[ 3QR = 48 \]

Divide both sides by 3:

\[ QR = \frac{48}{3} = 16 \]

So, the length of \( QR \) is \( 16 \) cm.

Would you like more details or have any questions?

Here are some related questions to expand on this topic:
1. How can you determine if two triangles are similar?
2. What other properties do similar triangles share?
3. How do you solve problems involving proportional segments in similar triangles?
4. Can the side lengths of similar triangles be used to find the ratio of their areas?
5. How does the concept of similarity help in solving real-world problems?

**Tip:** When working with proportions, always remember to cross-multiply to solve for the unknown variable.

Solve for QR when AB = 15 cm, BC = 12 cm, PQ = 20 cm, and the triangles ABC and PQR are proportional.

Discover how to determine the side lengths in similar triangles using proportions with a focus on triangles ABC and PQR where sides AB, BC, PQ, and QR are in a proportional relationship. Suitable for high school geometry, specifically Grades 9-10.

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