Key Concepts and Formulas

• Centroid of a Triangle: The centroid of a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ is given by $\left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right)$. The centroid is the point of intersection of the medians of the triangle.

• Equal Area Property of Centroid: The centroid divides a triangle into three smaller triangles of equal area.

• Distance Formula: The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

Step-by-Step Solution

1. Identify P as the Centroid

The problem states that the areas of triangles APC, APB, and BPC are equal. This implies that P is the centroid of triangle ABC.

• Why this step? Recognizing this property is crucial for solving the problem efficiently. The centroid is the only point inside a triangle that divides it into three triangles of equal area when connected to the vertices.

2. Calculate the Coordinates of the Centroid P

Given the vertices of $\triangle ABC$: $A(1, 0)$, $B(6, 2)$, and $C\left( \frac{3}{2}, 6 \right)$. Using the centroid formula: $P = \left( \frac{1 + 6 + \frac{3}{2}}{3}, \frac{0 + 2 + 6}{3} \right)$

• Why this step? We need to find the coordinates of P to calculate the distance PQ. Using the centroid formula is the direct method to find P's coordinates.

First, calculate the x-coordinate of P: $x_P = \frac{1 + 6 + \frac{3}{2}}{3} = \frac{\frac{2}{2} + \frac{12}{2} + \frac{3}{2}}{3} = \frac{\frac{17}{2}}{3} = \frac{17}{6}$

Next, calculate the y-coordinate of P: $y_P = \frac{0 + 2 + 6}{3} = \frac{8}{3}$

Therefore, the coordinates of point P are $\left( \frac{17}{6}, \frac{8}{3} \right)$.

3. State the Coordinates of Point Q

The problem provides the coordinates of point Q as: $Q = \left( -\frac{7}{6}, -\frac{1}{3} \right)$

• Why this step? We need the coordinates of both P and Q to calculate the distance PQ.

4. Calculate the Length of the Line Segment PQ

Using the distance formula with $P\left( \frac{17}{6}, \frac{8}{3} \right)$ and $Q\left( -\frac{7}{6}, -\frac{1}{3} \right)$: $PQ = \sqrt{\left( -\frac{7}{6} - \frac{17}{6} \right)^2 + \left( -\frac{1}{3} - \frac{8}{3} \right)^2}$

• Why this step? We apply the distance formula, which is the standard way to calculate the length of a line segment given the coordinates of its endpoints.

First, find the difference in x-coordinates: $x_Q - x_P = -\frac{7}{6} - \frac{17}{6} = -\frac{24}{6} = -4$

Next, find the difference in y-coordinates: $y_Q - y_P = -\frac{1}{3} - \frac{8}{3} = -\frac{9}{3} = -3$

Now, substitute these differences into the distance formula: $PQ = \sqrt{(-4)^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5$

The length of the line segment PQ is 5.

Common Mistakes & Tips

• Arithmetic Errors: Be careful with fractions and negative signs when calculating the centroid and applying the distance formula. Double-check your calculations.
• Misunderstanding the Centroid Property: If you don't recognize that P is the centroid, you'll waste time trying to calculate the areas of the smaller triangles directly, which is much more complicated.
• Incorrect Formula Application: Ensure you correctly apply the centroid and distance formulas.

Summary

The key to solving this problem was recognizing that the point P, which divides the triangle ABC into three triangles of equal area, is the centroid of the triangle. We calculated the coordinates of the centroid using the centroid formula and then used the distance formula to find the length of the line segment PQ, which is 5.

The final answer is \boxed{5}.