Let `DeltaABC -DeltaDEF , ar(DeltaABC)= 169 cm^(2) and ar (DeltaDEF) = 121 cm^(2)`. If AB = 26 cm thhen find DE.

Answer

To solve the problem, we will use the property of similar triangles, which states that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

### Step-by-step Solution:

1. **Identify the areas of the triangles:**
- Area of triangle ABC, \( \text{ar}(\Delta ABC) = 169 \, \text{cm}^2 \)
- Area of triangle DEF, \( \text{ar}(\Delta DEF) = 121 \, \text{cm}^2 \)

2. **Set up the ratio of the areas:**
\[ \frac{\text{ar}(\Delta ABC)}{\text{ar}(\Delta DEF)} = \frac{169}{121} \]

3. **Express the ratio of the areas in terms of the sides:**
Since the triangles are similar, we can write:
\[ \frac{\text{ar}(\Delta ABC)}{\text{ar}(\Delta DEF)} = \left(\frac{AB}{DE}\right)^2 \]
Therefore,
\[ \frac{169}{121} = \left(\frac{26}{DE}\right)^2 \]

4. **Take the square root of both sides:**
\[ \frac{13}{11} = \frac{26}{DE} \]

5. **Cross-multiply to solve for DE:**
\[ 13 \cdot DE = 11 \cdot 26 \]
\[ 13 \cdot DE = 286 \]

6. **Divide both sides by 13:**
\[ DE = \frac{286}{13} \]
\[ DE = 22 \, \text{cm} \]

### Final Answer:
The length of \( DE \) is \( 22 \, \text{cm} \).