Triangle ABC and DEF are similar. If AC = 19 cm and DF = 8 cm , the ratio of the area of two triangles are :

Options

A. `19/8`
B. `361/64`
C. `38/65`
D. None of these

Answer

To find the ratio of the areas of two similar triangles ABC and DEF, we can follow these steps:

### Step 1: Understand the properties of 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: Identify the corresponding sides
We are given:
- AC = 19 cm (side of triangle ABC)
- DF = 8 cm (corresponding side of triangle DEF)

### Step 3: Write the ratio of the sides
The ratio of the corresponding sides AC and DF can be written as:
\[ \text{Ratio of sides} = \frac{AC}{DF} = \frac{19}{8} \]

### Step 4: Square the ratio of the sides
To find the ratio of the areas, we need to square the ratio of the sides:
\[ \text{Ratio of areas} = \left(\frac{AC}{DF}\right)^2 = \left(\frac{19}{8}\right)^2 \]

### Step 5: Calculate the square of the ratio
Calculating the square:
\[ \left(\frac{19}{8}\right)^2 = \frac{19^2}{8^2} = \frac{361}{64} \]

### Step 6: State the final ratio of the areas
Thus, the ratio of the areas of triangle ABC to triangle DEF is:
\[ \text{Area of triangle ABC} : \text{Area of triangle DEF} = 361 : 64 \]