Let ` A ( 1, 0 ) , B ( 6, 2 ) ` and ` C (( 3 ) /(2), 6 ) ` be the vertices of a triangle ABC. If P is a point inside the triangle ABC such that the triangles APC, APB and BPC have equal areas , then the length of the line segment PQ, where Q is the point ` ( - ( 7 ) /(6), - ( 1 ) /(3)) , ` is _________.

Answer

To solve the problem step by step, we need to find the coordinates of point P, which is the centroid of triangle ABC, and then calculate the length of the line segment PQ where Q is given.

### Step 1: Find the coordinates of point P (Centroid of triangle ABC)

The vertices of triangle ABC are:
- A(1, 0)
- B(6, 2)
- C(3/2, 6)

The formula for the centroid (P) of a triangle with vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) is given by:
\[ P = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) \]

Substituting the coordinates of points A, B, and C:
\[ P_x = \frac{1 + 6 + \frac{3}{2}}{3} = \frac{1 + 6 + 1.5}{3} = \frac{8.5}{3} = \frac{17}{6} \]
\[ P_y = \frac{0 + 2 + 6}{3} = \frac{8}{3} \]

Thus, the coordinates of point P are:
\[ P\left(\frac{17}{6}, \frac{8}{3}\right) \]

### Step 2: Identify the coordinates of point Q

The coordinates of point Q are given as:
\[ Q\left(-\frac{7}{6}, -\frac{1}{3}\right) \]

### Step 3: Calculate the length of line segment PQ

To find the length of the line segment PQ, we use the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
where \((x_1, y_1)\) are the coordinates of point P and \((x_2, y_2)\) are the coordinates of point Q.

Substituting the coordinates of P and Q:
\[ d = \sqrt{\left(-\frac{7}{6} - \frac{17}{6}\right)^2 + \left(-\frac{1}{3} - \frac{8}{3}\right)^2} \]
\[ = \sqrt{\left(-\frac{24}{6}\right)^2 + \left(-\frac{9}{3}\right)^2} \]
\[ = \sqrt{(-4)^2 + (-3)^2} \]
\[ = \sqrt{16 + 9} \]
\[ = \sqrt{25} \]
\[ = 5 \]

### Final Answer

The length of the line segment PQ is:
\[ \boxed{5} \]