Find the area of the triangle formed by joining the mid-points of the sides of the triangle whose vertices are `(0,-1),(2,1)and (0,3).` Find the ratio of this area to the area of the given triangle.

Answer

To solve the problem, we will follow these steps:

### Step 1: Find the midpoints of the sides of the triangle.
The vertices of the triangle are given as \( A(0, -1) \), \( B(2, 1) \), and \( C(0, 3) \).

1. **Midpoint D of side AB**:
\[ D = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{0 + 2}{2}, \frac{-1 + 1}{2} \right) = \left( 1, 0 \right) \]

2. **Midpoint E of side BC**:
\[ E = \left( \frac{x_2 + x_3}{2}, \frac{y_2 + y_3}{2} \right) = \left( \frac{2 + 0}{2}, \frac{1 + 3}{2} \right) = \left( 1, 2 \right) \]

3. **Midpoint F of side CA**:
\[ F = \left( \frac{x_3 + x_1}{2}, \frac{y_3 + y_1}{2} \right) = \left( \frac{0 + 0}{2}, \frac{3 - 1}{2} \right) = \left( 0, 1 \right) \]

### Step 2: Calculate the area of triangle DEF.
Using the formula for the area of a triangle given vertices \( (x_1, y_1) \), \( (x_2, y_2) \), \( (x_3, y_3) \):
\[ \text{Area} = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right| \]
Substituting the coordinates of points D(1, 0), E(1, 2), and F(0, 1):
\[ \text{Area}_{DEF} = \frac{1}{2} \left| 1(2-1) + 1(1-0) + 0(0-2) \right| \]
Calculating inside the absolute value:
\[ = \frac{1}{2} \left| 1(1) + 1(1) + 0 \right| = \frac{1}{2} \left| 1 + 1 \right| = \frac{1}{2} \times 2 = 1 \]

### Step 3: Calculate the area of triangle ABC.
Using the same area formula for triangle ABC with vertices A(0, -1), B(2, 1), and C(0, 3):
\[ \text{Area}_{ABC} = \frac{1}{2} \left| 0(1-3) + 2(3+1) + 0(-1-1) \right| \]
Calculating inside the absolute value:
\[ = \frac{1}{2} \left| 0 + 2(4) + 0 \right| = \frac{1}{2} \left| 8 \right| = \frac{8}{2} = 4 \]

### Step 4: Find the ratio of the areas.
The ratio of the area of triangle DEF to the area of triangle ABC is:
\[ \text{Ratio} = \frac{\text{Area}_{DEF}}{\text{Area}_{ABC}} = \frac{1}{4} \]

### Final Answer:
The area of triangle DEF is \( 1 \) square unit, and the ratio of the area of triangle DEF to the area of triangle ABC is \( 1:4 \).

---