To find the length of the line segment \( PB \) where \( P \) is a point inside triangle \( ABC \) such that the areas of triangles \( APC \), \( APB \), and \( BPC \) are equal, we can follow these steps:
### Step 1: Identify the coordinates of the vertices
The vertices of triangle \( ABC \) are given as:
- \( A(5, -3) \)
- \( B(2, 7) \)
- \( C(-1, 2) \)
### Step 2: Use the property of equal areas
Since the areas of triangles \( APC \), \( APB \), and \( BPC \) are equal, point \( P \) must be the centroid of triangle \( ABC \). The centroid \( G \) of a triangle with vertices \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) is given by the formula:
\[
G\left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right)
\]
### Step 3: Calculate the coordinates of the centroid \( P \)
Using the coordinates of points \( A \), \( B \), and \( C \):
\[
x_P = \frac{5 + 2 - 1}{3} = \frac{6}{3} = 2
\]
\[
y_P = \frac{-3 + 7 + 2}{3} = \frac{6}{3} = 2
\]
Thus, the coordinates of point \( P \) are \( P(2, 2) \).
### Step 4: Use the distance formula to find \( PB \)
The distance \( PB \) can be calculated using the distance formula:
\[
PB = \sqrt{(x_B - x_P)^2 + (y_B - y_P)^2}
\]
Substituting the coordinates of \( P \) and \( B \):
- \( B(2, 7) \)
- \( P(2, 2) \)
Calculating:
\[
PB = \sqrt{(2 - 2)^2 + (7 - 2)^2} = \sqrt{0 + 5^2} = \sqrt{25} = 5
\]
### Final Answer
The length of the line segment \( PB \) is \( 5 \) units.
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