If `DeltaABC` and `DeltaDEF` are two tnangles such that `(AB)/(EF)=(BC)/(FD)=(CA)/(DE)=3/4` then find `ar(DeltaDEF): ar(DeltaABC)`.

Answer

To find the ratio of the areas of triangles \( \Delta DEF \) and \( \Delta ABC \), we can follow these steps:

### Step 1: Understand the given ratios
We know that:
\[ \frac{AB}{EF} = \frac{BC}{FD} = \frac{CA}{DE} = \frac{3}{4} \]
This means that the sides of triangle \( ABC \) are proportional to the sides of triangle \( DEF \) with a ratio of \( \frac{3}{4} \).

### Step 2: Establish similarity of triangles
Since the ratios of corresponding sides are equal, we can conclude that triangle \( ABC \) is similar to triangle \( DEF \) by the Side-Side-Side (SSS) similarity criterion.

### Step 3: Use the property of similar triangles
For two similar triangles, the ratio of their areas is equal to the square of the ratio of their corresponding sides. Therefore, we can write:
\[ \frac{ar(\Delta DEF)}{ar(\Delta ABC)} = \left(\frac{EF}{AB}\right)^2 = \left(\frac{4}{3}\right)^2 \]

### Step 4: Calculate the ratio of areas
Now, squaring the ratio \( \frac{4}{3} \):
\[ \left(\frac{4}{3}\right)^2 = \frac{16}{9} \]
Thus, we have:
\[ \frac{ar(\Delta DEF)}{ar(\Delta ABC)} = \frac{16}{9} \]

### Conclusion
The ratio of the areas of triangles \( \Delta DEF \) and \( \Delta ABC \) is:
\[ ar(\Delta DEF) : ar(\Delta ABC) = 16 : 9 \]

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