Let `ABC ~ triangle DEF` and their areas be `81 cm^(2)` and `144 cm^(2)`. If EF = 24 cm, then length of side BC is ……………………. Cm

Answer

To find the length of side BC in triangle ABC given that triangle ABC is similar to triangle DEF, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between the areas of similar triangles**:
The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.
\[ \frac{\text{Area of } ABC}{\text{Area of } DEF} = \left(\frac{BC}{EF}\right)^2 \]

2. **Substitute the given areas**:
We know the areas of triangles ABC and DEF:
\[ \frac{81 \, \text{cm}^2}{144 \, \text{cm}^2} = \left(\frac{BC}{24 \, \text{cm}}\right)^2 \]

3. **Simplify the area ratio**:
Simplifying \(\frac{81}{144}\):
\[ \frac{81}{144} = \frac{9}{16} \]
So we have:
\[ \frac{9}{16} = \left(\frac{BC}{24}\right)^2 \]

4. **Take the square root of both sides**:
Taking the square root gives us:
\[ \frac{3}{4} = \frac{BC}{24} \]

5. **Cross-multiply to solve for BC**:
Cross-multiplying gives:
\[ 3 \times 24 = 4 \times BC \]
\[ 72 = 4 \times BC \]

6. **Divide by 4**:
To isolate BC, divide both sides by 4:
\[ BC = \frac{72}{4} = 18 \, \text{cm} \]

### Final Answer:
The length of side BC is **18 cm**.

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