Which one of the following statements is true about the line segment with endpoints `(-1,1)` and `(1, -1)` ?

Options

A. Crosses the x-axis only
B. Crosses the y-axis only
C. Crosses the y -axis on its positive side
D. Passes through the origin (0,0)

Answer

To determine which statement is true about the line segment with endpoints \((-1, 1)\) and \((1, -1)\), we will follow these steps:

### Step 1: Identify the Points
Let the endpoints of the line segment be:
- Point 1: \( (x_1, y_1) = (-1, 1) \)
- Point 2: \( (x_2, y_2) = (1, -1) \)

### Step 2: Calculate the Slope of the Line
The slope \( m \) of the line joining two points is calculated using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the values:
\[ m = \frac{-1 - 1}{1 - (-1)} = \frac{-2}{2} = -1 \]

### Step 3: Write the Equation of the Line
Using the point-slope form of the equation of a line:
\[ y - y_1 = m(x - x_1) \]
Substituting \( m = -1 \), \( x_1 = -1 \), and \( y_1 = 1 \):
\[ y - 1 = -1(x + 1) \]
Simplifying this:
\[ y - 1 = -x - 1 \]
\[ y = -x \]

### Step 4: Check if the Line Passes Through the Origin
We need to check if the line \( y = -x \) passes through the origin \( (0, 0) \):
Substituting \( x = 0 \):
\[ y = -0 = 0 \]
Thus, the point \( (0, 0) \) satisfies the equation, confirming that the line passes through the origin.

### Step 5: Determine the Implications
Since the line passes through the origin, it does not intersect the x-axis or y-axis at any other point besides the origin.

### Conclusion
The correct statement regarding the line segment with endpoints \((-1, 1)\) and \((1, -1)\) is that it passes through the origin and does not intersect the axes at any other points.