The coordinates of A, B and C are (–1, 5), (3, 1) and (5, 7) respectively, D, E and F are the middle points of BC, CA and AB respectively. Calculate the area of the triangle DEF.
(a)4
(b)3
(c)2
(d)8

Answer

To find the area of triangle DEF, we first need to determine the coordinates of points D, E, and F, which are the midpoints of sides BC, CA, and AB of triangle ABC respectively. The coordinates of points A, B, and C are given as follows:

- A = (-1, 5)
- B = (3, 1)
- C = (5, 7)

### Step 1: Calculate the midpoints D, E, and F

1. **Midpoint D (of BC)**:
\[ D = \left( \frac{x_B + x_C}{2}, \frac{y_B + y_C}{2} \right) = \left( \frac{3 + 5}{2}, \frac{1 + 7}{2} \right) = \left( \frac{8}{2}, \frac{8}{2} \right) = (4, 4) \]

2. **Midpoint E (of CA)**:
\[ E = \left( \frac{x_C + x_A}{2}, \frac{y_C + y_A}{2} \right) = \left( \frac{5 + (-1)}{2}, \frac{7 + 5}{2} \right) = \left( \frac{4}{2}, \frac{12}{2} \right) = (2, 6) \]

3. **Midpoint F (of AB)**:
\[ F = \left( \frac{x_A + x_B}{2}, \frac{y_A + y_B}{2} \right) = \left( \frac{-1 + 3}{2}, \frac{5 + 1}{2} \right) = \left( \frac{2}{2}, \frac{6}{2} \right) = (1, 3) \]

Now we have the coordinates of points D, E, and F:
- D = (4, 4)
- E = (2, 6)
- F = (1, 3)

### Step 2: Calculate the area of triangle DEF

We can use the formula for the area of a triangle given its vertices (x1, y1), (x2, y2), (x3, y3):
\[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]

Substituting the coordinates of D, E, and F:
\[ \text{Area} = \frac{1}{2} \left| 4(6 - 3) + 2(3 - 4) + 1(4 - 6) \right| \]
\[ = \frac{1}{2} \left| 4 \cdot 3 + 2 \cdot (-1) + 1 \cdot (-2) \right| \]
\[ = \frac{1}{2} \left| 12 - 2 - 2 \right| \]
\[ = \frac{1}{2} \left| 8 \right| = \frac{8}{2} = 4 \]

### Conclusion
The area of triangle DEF is 4.

### Final Answer
(a) 4