Question

The mid-point of the line segment joining the points (5, -4) and (6, 4) lies on :

• A. x-axis
• B. y-axis
• C. origin
• D. neither x-axis nor y-axis

Hint:

If two points have y-coordinates that are negatives of each other (like -4 and 4), their midpoint will always have a y-coordinate of 0 and thus lie on the x-axis.

Answer 1

The Correct Option is A

To determine whether the midpoint of the line segment joining the points \( (5, -4) \) and \( (6, 4) \) lies on the x-axis, y-axis, origin, or neither, we can use the formula for finding the midpoint of a line segment.

Midpoint Formula:

The midpoint \((M)\) of a line segment joining two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula:

\(M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\)

Calculation:

Here, the points are \((5, -4)\) and \((6, 4)\). Substituting these into the formula, we get:

• \(x_{\text{mid}} = \frac{5 + 6}{2} = \frac{11}{2} = 5.5\)
• \(y_{\text{mid}} = \frac{-4 + 4}{2} = \frac{0}{2} = 0\)

So, the midpoint is \((5.5, 0)\).

Conclusion:

The coordinate \(y = 0\) indicates that the point lies on the x-axis. Therefore, the midpoint \((5.5, 0)\) lies on the x-axis.

Verification:

• x-axis: Points on the x-axis have \(y = 0\). Since the midpoint's y-coordinate is 0, it is on the x-axis.
• y-axis: Points on the y-axis have \(x = 0\). This is not the case, as \(x_{\text{mid}} = 5.5\).
• Origin: The origin is at (0, 0). This does not match the midpoint.
• Neither: Since the midpoint lies on the x-axis, this option is incorrect.

Therefore, the correct answer is that the midpoint lies on the x-axis.