**Given Information:**
- The vertices of triangle ABC are A(4,-6), B(3,-2), and C(5,2).
- We need to verify that the median of triangle ABC divides it into two triangles of equal areas.

**Step 1: Find the coordinates of the midpoint of side AB.**

The midpoint of a line segment with endpoints (x₁, y₁) and (x₂, y₂) can be found using the midpoint formula:

Midpoint = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)

Using this formula, we can find the coordinates of the midpoint of side AB:

Midpoint of AB = ((4 + 3) / 2, (-6 + (-2)) / 2)
= (7/2, -8/2)
= (7/2, -4)

So, the midpoint of side AB is M(7/2, -4).

**Step 2: Find the equation of the line passing through points C and M.**

The equation of a line passing through two points (x₁, y₁) and (x₂, y₂) can be found using the point-slope form:

y - y₁ = m(x - x₁)

where m is the slope of the line.

The slope of the line passing through points C(5,2) and M(7/2, -4) can be found using the slope formula:

m = (y₂ - y₁) / (x₂ - x₁)

m = (-4 - 2) / (7/2 - 5)
= -6 / (-3/2)
= -6 * (-2/3)
= 4

So, the equation of the line passing through points C and M is:

y - 2 = 4(x - 5)

Simplifying the equation, we get:

y - 2 = 4x - 20
y = 4x - 18

**Step 3: Find the coordinates of the intersection point of the median and the line.**

To find the intersection point, we need to solve the system of equations formed by the line equation and the equation of the median passing through point C.

The equation of the median passing through point C can be found using the point-slope form:

y - y₁ = m(x - x₁)

where (x₁, y₁) is the coordinate of point C and m is the slope of the median. Since the median passes through point C(5,2) and the midpoint of side AB, the slope of the median is the negative reciprocal of the slope of the line passing through points C and M.

The slope of the median = -1 / 4

Using the point-slope form, we have:

y - 2 = (-1/4)(x - 5)
y - 2 = (-1/4)x + 5/4

Simplifying the equation, we get:

y = (-1/4)x + 23/4

Now, we can solve the system of equations:

4x - 18 = (-1/4)x + 23/4

Multiplying both sides by 4 to eliminate the fractions:

16x - 72 = -x + 23

Bringing like terms to one