Find the area of the triangle formed by joining the mid -points of the sides of the triangle ABC ,whose vertices are A(0,-1),B(2,1)andC(0,3).

Answer

To find the area of the triangle formed by joining the midpoints of the sides of triangle ABC with vertices A(0, -1), B(2, 1), and C(0, 3), we will follow these steps:

### Step 1: Find the midpoints of the sides of triangle ABC

1. **Midpoint of AB (let's call it P)**:
- Coordinates of A = (0, -1)
- Coordinates of B = (2, 1)
- Formula for midpoint: \( P = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \)
- Calculation:
\[ P = \left( \frac{0 + 2}{2}, \frac{-1 + 1}{2} \right) = \left( 1, 0 \right) \]

2. **Midpoint of AC (let's call it Q)**:
- Coordinates of A = (0, -1)
- Coordinates of C = (0, 3)
- Calculation:
\[ Q = \left( \frac{0 + 0}{2}, \frac{-1 + 3}{2} \right) = \left( 0, 1 \right) \]

3. **Midpoint of BC (let's call it R)**:
- Coordinates of B = (2, 1)
- Coordinates of C = (0, 3)
- Calculation:
\[ R = \left( \frac{2 + 0}{2}, \frac{1 + 3}{2} \right) = \left( 1, 2 \right) \]

### Step 2: Use the coordinates of P, Q, and R to find the area of triangle PQR

The coordinates of the midpoints are:
- P(1, 0)
- Q(0, 1)
- R(1, 2)

We will use the formula for the area of a triangle given by vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\):
\[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]

Substituting the coordinates:
- \(x_1 = 1\), \(y_1 = 0\)
- \(x_2 = 0\), \(y_2 = 1\)
- \(x_3 = 1\), \(y_3 = 2\)

### Step 3: Calculate the area

Substituting into the area formula:
\[ \text{Area} = \frac{1}{2} \left| 1(1 - 2) + 0(2 - 0) + 1(0 - 1) \right| \]
Calculating each term:
\[ = \frac{1}{2} \left| 1(-1) + 0 + 1(-1) \right| \]
\[ = \frac{1}{2} \left| -1 + 0 - 1 \right| \]
\[ = \frac{1}{2} \left| -2 \right| = \frac{1}{2} \times 2 = 1 \]

### Final Answer
The area of triangle PQR is \(1\) square unit.

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