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Which one of the following statements is...

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

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)

Text Solution

AI Generated Solution

The correct Answer is:
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.
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