If ` A B C ∼ D E F` such that `A B=1. 2 c m` and `D E=1. 4 c m` . Find the ratio of areas of ` A B C` and ` D E F` .

Answer

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

### Step 1: Understand the relationship between similar 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 information
We have:
- Length of side \( AB = 1.2 \, \text{cm} \)
- Length of side \( DE = 1.4 \, \text{cm} \)

### Step 3: Set up the ratio of the sides
The ratio of the sides \( AB \) and \( DE \) can be expressed as:
\[ \frac{AB}{DE} = \frac{1.2}{1.4} \]

### Step 4: Simplify the ratio
To simplify \( \frac{1.2}{1.4} \), we can multiply both the numerator and the denominator by 10 to eliminate the decimal:
\[ \frac{1.2 \times 10}{1.4 \times 10} = \frac{12}{14} \]
Now, we can simplify \( \frac{12}{14} \) by dividing both the numerator and the denominator by 2:
\[ \frac{12 \div 2}{14 \div 2} = \frac{6}{7} \]

### Step 5: Calculate the ratio of the areas
The ratio of the areas of triangles \( ABC \) and \( DEF \) is given by the square of the ratio of the sides:
\[ \frac{\text{Area of } ABC}{\text{Area of } DEF} = \left(\frac{AB}{DE}\right)^2 = \left(\frac{6}{7}\right)^2 \]

### Step 6: Compute the square of the ratio
Calculating \( \left(\frac{6}{7}\right)^2 \):
\[ \left(\frac{6}{7}\right)^2 = \frac{6^2}{7^2} = \frac{36}{49} \]

### Step 7: State the final answer
Thus, the ratio of the areas of triangles \( ABC \) and \( DEF \) is:
\[ \frac{\text{Area of } ABC}{\text{Area of } DEF} = \frac{36}{49} \]