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 will first find the coordinates of point P, which is the centroid of triangle ABC, and then calculate the distance between points P and Q.

### Step 1: Find the coordinates of the centroid P 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 (x1, y1), (x2, y2), and (x3, y3) 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 A, B, and C:
\[ P\left(\frac{1 + 6 + \frac{3}{2}}{3}, \frac{0 + 2 + 6}{3}\right) \]

Calculating the x-coordinate:
\[ x_P = \frac{1 + 6 + \frac{3}{2}}{3} = \frac{7 + \frac{3}{2}}{3} = \frac{\frac{14}{2} + \frac{3}{2}}{3} = \frac{\frac{17}{2}}{3} = \frac{17}{6} \]

Calculating the y-coordinate:
\[ y_P = \frac{0 + 2 + 6}{3} = \frac{8}{3} \]

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 distance PQ using the distance formula.

The distance \(d\) between two points \(P(x_1, y_1)\) and \(Q(x_2, y_2)\) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

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

Calculating the x-component:
\[ -\frac{7}{6} - \frac{17}{6} = -\frac{24}{6} = -4 \]

Calculating the y-component:
\[ -\frac{1}{3} - \frac{8}{3} = -\frac{9}{3} = -3 \]

Thus, we have:
\[ d = \sqrt{(-4)^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]

### Final Answer:
The length of the line segment PQ is \(5\).

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