ABC is a triangle and D,E,F are the mid-points of the sides BC,CA, AB respectively . The ratio of the areas of `DeltaABC and DeltaDEF` is :

Options

A. `4:1`
B. `5:1`
C. `3:1`
D. can't be determined

Answer

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

### Step 1: Identify the midpoints
Let triangle ABC have vertices A, B, and C. The midpoints of the sides BC, CA, and AB are D, E, and F, respectively.

### Step 2: Understand the properties of midpoints
Since D, E, and F are midpoints, the segments AD, BE, and CF will each divide the triangle into smaller triangles of equal area. Each of these smaller triangles (like ADF, BEF, and CDE) will have an area that is a fraction of the area of triangle ABC.

### Step 3: Calculate the area ratio
The area of triangle DEF can be derived from the area of triangle ABC. The area of triangle DEF is exactly one-fourth of the area of triangle ABC. This is because the triangles formed by connecting the midpoints of a triangle (like DEF) have an area that is a quarter of the area of the original triangle (ABC).

### Step 4: Set up the ratio
Let the area of triangle ABC be denoted as Area(ABC) and the area of triangle DEF be denoted as Area(DEF). We have:
- Area(ABC) = 4 * Area(DEF)

### Step 5: Express the ratio
Thus, the ratio of the areas of triangle ABC to triangle DEF is:
\[ \text{Ratio} = \frac{\text{Area(ABC)}}{\text{Area(DEF)}} = \frac{4}{1} \]

### Final Answer
The ratio of the areas of triangle ABC to triangle DEF is 4:1.

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