Given that ΔABC∼ΔDEF,BC=3cm,EF=4cm ans the area of ΔABC is 54sq.cm.Then find the area of ΔDEF.

Answer

To find the area of triangle DEF given that triangle ABC is similar to triangle DEF, we can follow these steps:

### Step 1: Understand the relationship between the triangles
Since triangles ABC and DEF are similar, the ratio of their areas is equal to the square of the ratio of their corresponding sides.

### Step 2: Write down the given values
- Length of side BC in triangle ABC = 3 cm
- Length of side EF in triangle DEF = 4 cm
- Area of triangle ABC = 54 sq. cm

### Step 3: Set up the ratio of the sides
The ratio of the sides BC and EF is:
\[ \frac{BC}{EF} = \frac{3}{4} \]

### Step 4: Set up the ratio of the areas
Since the triangles are similar, the ratio of their areas is given by:
\[ \frac{\text{Area of } \triangle ABC}{\text{Area of } \triangle DEF} = \left(\frac{BC}{EF}\right)^2 \]
Substituting the known values:
\[ \frac{54}{\text{Area of } \triangle DEF} = \left(\frac{3}{4}\right)^2 \]

### Step 5: Calculate the square of the ratio of the sides
Calculating \(\left(\frac{3}{4}\right)^2\):
\[ \left(\frac{3}{4}\right)^2 = \frac{9}{16} \]

### Step 6: Set up the equation
Now we can set up the equation:
\[ \frac{54}{\text{Area of } \triangle DEF} = \frac{9}{16} \]

### Step 7: Cross-multiply to solve for the area of triangle DEF
Cross-multiplying gives us:
\[ 54 \cdot 16 = 9 \cdot \text{Area of } \triangle DEF \]
Calculating \(54 \cdot 16\):
\[ 54 \cdot 16 = 864 \]
So,
\[ 864 = 9 \cdot \text{Area of } \triangle DEF \]

### Step 8: Solve for the area of triangle DEF
Now divide both sides by 9:
\[ \text{Area of } \triangle DEF = \frac{864}{9} = 96 \text{ sq. cm} \]

### Final Answer
The area of triangle DEF is \(96 \text{ sq. cm}\).

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