The perimeters of two similar triangles are 26 cm and 39 cm.The ratio of their areas will be:

Options

A. `2:3`
B. `6:9`
C. `4:6`
D. `4:9`

Answer

To find the ratio of the areas of two similar triangles given their perimeters, we can follow these steps:

### Step 1: Identify the ratio of the perimeters
The perimeters of the two triangles are given as 26 cm and 39 cm. To find the ratio of the perimeters, we divide the smaller perimeter by the larger perimeter.

\[ \text{Ratio of perimeters} = \frac{26}{39} \]

### Step 2: Simplify the ratio of the perimeters
Now, we simplify the ratio:

\[ \frac{26}{39} = \frac{2}{3} \]

### Step 3: Find the ratio of the areas
For similar triangles, the ratio of their areas is equal to the square of the ratio of their corresponding sides (which is the same as the ratio of their perimeters). Therefore, we square the ratio we found in Step 2:

\[ \text{Ratio of areas} = \left(\frac{2}{3}\right)^2 = \frac{2^2}{3^2} = \frac{4}{9} \]

### Final Answer
Thus, the ratio of the areas of the two similar triangles is:

\[ \frac{4}{9} \]

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