As we know studied that a median of a triangle divides it into triangles of equal areas. Verify this result for `Delta ABC` whose vertices are A (4,-6), B(3,-2) and C (5,2).

Answer

To verify that the median of triangle ABC divides it into two triangles of equal area, we will follow these steps:

### Step 1: Identify the vertices of the triangle
The vertices of triangle ABC are given as:
- A (4, -6)
- B (3, -2)
- C (5, 2)

### Step 2: Find the midpoint D of side BC
To find the midpoint D of side BC, we use the midpoint formula:
\[ D\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \]
where \( B(3, -2) \) and \( C(5, 2) \).

Calculating the coordinates of D:
\[ D\left(\frac{3 + 5}{2}, \frac{-2 + 2}{2}\right) = D\left(\frac{8}{2}, \frac{0}{2}\right) = D(4, 0) \]

### Step 3: Calculate the area of triangle ABD
We will use the formula for the area of a triangle given by vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\):
\[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
For triangle ABD, the vertices are:
- A(4, -6)
- B(3, -2)
- D(4, 0)

Substituting the coordinates into the area formula:
\[ \text{Area}_{ABD} = \frac{1}{2} \left| 4(-2 - 0) + 3(0 + 6) + 4(-6 + 2) \right| \]
Calculating this:
\[ = \frac{1}{2} \left| 4(-2) + 3(6) + 4(-4) \right| \]
\[ = \frac{1}{2} \left| -8 + 18 - 16 \right| \]
\[ = \frac{1}{2} \left| -6 \right| = \frac{6}{2} = 3 \]

### Step 4: Calculate the area of triangle ADC
For triangle ADC, the vertices are:
- A(4, -6)
- D(4, 0)
- C(5, 2)

Using the area formula again:
\[ \text{Area}_{ADC} = \frac{1}{2} \left| 4(0 - 2) + 4(2 + 6) + 5(-6 - 0) \right| \]
Calculating this:
\[ = \frac{1}{2} \left| 4(-2) + 4(8) + 5(-6) \right| \]
\[ = \frac{1}{2} \left| -8 + 32 - 30 \right| \]
\[ = \frac{1}{2} \left| -6 \right| = \frac{6}{2} = 3 \]

### Step 5: Compare the areas
We have found that:
- Area of triangle ABD = 3
- Area of triangle ADC = 3

Since both areas are equal, we have verified that the median AD divides triangle ABC into two triangles of equal area.

### Conclusion
Thus, we can conclude that the median of a triangle divides it into two triangles of equal area. Hence proved.

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