Area of two similar triangles are `4" cm"^(2)` and `9" cm"^(2)` respectively . The ratio of their corresponding sides is

Options

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

Answer

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

### Step 1: Understand the relationship between the areas and the sides of similar triangles.
For two similar triangles, the ratio of their areas is equal to the square of the ratio of their corresponding sides.

### Step 2: Set up the equation based on the given areas.
Let the areas of the two triangles be \( A_1 = 4 \, \text{cm}^2 \) and \( A_2 = 9 \, \text{cm}^2 \). According to the property of similar triangles, we can write:
\[ \frac{A_1}{A_2} = \left(\frac{a}{b}\right)^2 \]
where \( a \) and \( b \) are the lengths of the corresponding sides of the triangles.

### Step 3: Substitute the values of the areas into the equation.
Substituting the values of the areas into the equation gives us:
\[ \frac{4}{9} = \left(\frac{a}{b}\right)^2 \]

### Step 4: Take the square root of both sides to find the ratio of the sides.
To find the ratio of the sides \( \frac{a}{b} \), we take the square root of both sides:
\[ \frac{a}{b} = \sqrt{\frac{4}{9}} = \frac{\sqrt{4}}{\sqrt{9}} = \frac{2}{3} \]

### Step 5: State the final answer.
Thus, the ratio of the corresponding sides of the two similar triangles is:
\[ \frac{a}{b} = 2:3 \]