The square with coordinates (5,1), (8,2), (7, 5) and (4,4) has its area bisected by the line L, if the line L passes through the origin, what is its equation?

To find the equation of line \( L \), which passes through the origin and bisects the area of the square with given vertices, we proceed with the following steps:

### Step 1: Find the center of the square
The center of a square can be found by averaging the coordinates of its vertices:
- Vertex 1: \( (5, 1) \)
- Vertex 2: \( (8, 2) \)
- Vertex 3: \( (7, 5) \)
- Vertex 4: \( (4, 4) \)

The center \( C \) is:
\[
C = \left( \frac{5+8+7+4}{4}, \frac{1+2+5+4}{4} \right) = \left( \frac{24}{4}, \frac{12}{4} \right) = (6, 3)
\]

### Step 2: Determine the slope of the line
Since the line passes through the origin and bisects the area of the square, it must also pass through the center of the square, \( (6, 3) \).

Using the point-slope form of a line, the slope \( m \) is given by:
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 0}{6 - 0} = \frac{3}{6} = \frac{1}{2}
\]

### Step 3: Equation of the line
The equation of the line with slope \( m = \frac{1}{2} \) and passing through the origin is:
\[
y = \frac{1}{2}x
\]

Thus, the equation of the line \( L \) is:
\[
y = \frac{1}{2}x
\]

### Conclusion:
The equation of the line that bisects the square is \( \boxed{y = \frac{1}{2}x} \).

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Do you want more details or have any questions? Here are 5 related questions to expand your understanding:

1. How can we calculate the area of this square based on its vertices?
2. How do we confirm that this line bisects the area of the square?
3. What is the distance between the origin and the center of the square?
4. How do we verify if a line passes through a given point?
5. Can we find another line that bisects the square and does not pass through the origin?

**Tip:** To find the slope between two points, always remember the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \).

Find the equation of a line that bisects the area of a square with vertices at (5,1), (8,2), (7,5), and (4,4) and passes through the origin. This problem involves concepts of coordinate geometry, line equations, and the slope formula, suitable for students in Grades 9-12.

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