`Delta DEF` is similar to `Delta GHI`. The length of DE is 9 cm and length of the corresponding side GH is 16 cm. What is the ratio of areas of `Delta DEF : Delta GHI` ?

Options

A. `3:4`
B. `9:16`
C. `81:256`
D. `16:9`

Answer

To find the ratio of the areas of triangle DEF to triangle GHI, we can follow these steps:

### Step 1: Understand the relationship between similar triangles
Since triangle DEF is similar to triangle GHI, the ratio of their areas is equal to the square of the ratio of their corresponding sides.

### Step 2: Identify the lengths of the corresponding sides
We are given:
- Length of side DE = 9 cm
- Length of corresponding side GH = 16 cm

### Step 3: Calculate the ratio of the corresponding sides
The ratio of the lengths of the sides DE to GH is:
\[ \text{Ratio of sides} = \frac{DE}{GH} = \frac{9 \text{ cm}}{16 \text{ cm}} = \frac{9}{16} \]

### Step 4: Calculate the ratio of the areas
The ratio of the areas of the triangles is the square of the ratio of the corresponding sides:
\[ \text{Ratio of areas} = \left(\frac{DE}{GH}\right)^2 = \left(\frac{9}{16}\right)^2 \]

### Step 5: Perform the squaring operation
Calculating the square of the ratio:
\[ \left(\frac{9}{16}\right)^2 = \frac{9^2}{16^2} = \frac{81}{256} \]

### Step 6: Write the final answer
Thus, the ratio of the areas of triangle DEF to triangle GHI is:
\[ \text{Area Ratio} = 81 : 256 \]

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