`DeltaABC~DeltaPQR`. If `AB=4cm`, `PQ=6cm` and `QR=9cm`. Find `BC`.

Options

A. 5 cm
B. 6 cm
C. 7 cm
D. 8 cm

Answer

To solve the problem, we will use the property of similar triangles which states that the ratios of the lengths of corresponding sides are equal.

**Step-by-step solution:**

1. **Identify the given information:**
- We have two similar triangles: ΔABC ~ ΔPQR.
- The lengths of the sides are given as:
- AB = 4 cm
- PQ = 6 cm
- QR = 9 cm
- We need to find the length of side BC.

2. **Set up the proportion using the similarity of triangles:**
Since the triangles are similar, we can write the proportion of the corresponding sides:
\[ \frac{AB}{PQ} = \frac{BC}{QR} \]

3. **Substitute the known values into the proportion:**
Substitute AB, PQ, and QR into the proportion:
\[ \frac{4}{6} = \frac{BC}{9} \]

4. **Cross-multiply to solve for BC:**
Cross-multiplying gives us:
\[ 4 \cdot 9 = 6 \cdot BC \]
This simplifies to:
\[ 36 = 6 \cdot BC \]

5. **Isolate BC:**
To find BC, divide both sides by 6:
\[ BC = \frac{36}{6} = 6 \text{ cm} \]

6. **Conclusion:**
The length of side BC is 6 cm.