Find the area of the triangle formed by joining the mid-points of the sides of the triangle, whose vertices are (2, - 4), (6, 2) and (- 4, 6). Find the ratio of this area to the area of the given triangle.

Answer

To find the area of the triangle formed by joining the midpoints of the sides of the triangle with vertices at (2, -4), (6, 2), and (-4, 6), and to find the ratio of this area to the area of the given triangle, we can follow these steps:

### Step 1: Calculate the area of the original triangle (ABC)

The formula for the area of a triangle given its vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) is:

\[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]

Substituting the coordinates of the vertices:
- \(A(2, -4)\)
- \(B(6, 2)\)
- \(C(-4, 6)\)

We have:
\[ \text{Area} = \frac{1}{2} \left| 2(2 - 6) + 6(6 + 4) + (-4)(-4 - 2) \right| \]

Calculating each term:
- \(2(2 - 6) = 2 \times (-4) = -8\)
- \(6(6 + 4) = 6 \times 10 = 60\)
- \((-4)(-4 - 2) = -4 \times (-6) = 24\)

Now substituting these back into the area formula:
\[ \text{Area} = \frac{1}{2} \left| -8 + 60 + 24 \right| = \frac{1}{2} \left| 76 \right| = \frac{76}{2} = 38 \text{ square units} \]

### Step 2: Find the midpoints of the sides of triangle ABC

The midpoints of the sides can be calculated using the midpoint formula:
\[ \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

1. Midpoint \(P\) of \(AB\):
\[ P = \left( \frac{2 + 6}{2}, \frac{-4 + 2}{2} \right) = \left( \frac{8}{2}, \frac{-2}{2} \right) = (4, -1) \]

2. Midpoint \(Q\) of \(BC\):
\[ Q = \left( \frac{6 + (-4)}{2}, \frac{2 + 6}{2} \right) = \left( \frac{2}{2}, \frac{8}{2} \right) = (1, 4) \]

3. Midpoint \(R\) of \(CA\):
\[ R = \left( \frac{-4 + 2}{2}, \frac{6 + (-4)}{2} \right) = \left( \frac{-2}{2}, \frac{2}{2} \right) = (-1, 1) \]

### Step 3: Calculate the area of triangle PQR

Using the same area formula for triangle PQR:
\[ \text{Area}_{PQR} = \frac{1}{2} \left| 4(4 - 1) + 1(1 + 1) + (-1)(-1 - 4) \right| \]

Calculating each term:
- \(4(4 - 1) = 4 \times 3 = 12\)
- \(1(1 + 1) = 1 \times 2 = 2\)
- \((-1)(-1 - 4) = -1 \times (-5) = 5\)

Now substituting these back into the area formula:
\[ \text{Area}_{PQR} = \frac{1}{2} \left| 12 + 2 + 5 \right| = \frac{1}{2} \left| 19 \right| = \frac{19}{2} = 9.5 \text{ square units} \]

### Step 4: Find the ratio of the areas

The ratio of the area of triangle PQR to the area of triangle ABC is:
\[ \text{Ratio} = \frac{\text{Area}_{PQR}}{\text{Area}_{ABC}} = \frac{9.5}{38} = \frac{1}{4} \]

### Final Answer:
- Area of triangle ABC: \(38\) square units
- Area of triangle PQR: \(9.5\) square units
- Ratio of areas: \(\frac{1}{4}\)