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 you see two \(y\)-coordinates that are negatives of each other (like -4 and 4), their average will always be 0, meaning the midpoint will always lie on the \(x\)-axis (provided they are not both zero).

Answer 1

The Correct Option is A

The given problem requires finding the mid-point of a line segment and deducing its position in relation to the coordinate axes. Let's go through the problem step-by-step:

Step 1: Identify the Points and Use the Mid-Point Formula

• The given points are \((5, -4)\) and \((6, 4)\).
• The mid-point formula for the line segment joining two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

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

Step 2: Calculate the Mid-Point

• Substitute the given points into the formula:

\(\left( \frac{5 + 6}{2}, \frac{-4 + 4}{2} \right)\)

• Simplifying, we get:

\(\left( \frac{11}{2}, 0 \right)\)

Step 3: Determine the Position of the Mid-Point

• The mid-point is \(\left( \frac{11}{2}, 0 \right)\), which clearly has its \(y\)-coordinate as 0.
• In the coordinate plane, a point with \(y = 0\) lies on the \(x\)-axis.

Conclusion: Therefore, the mid-point of the line segment joining the points \((5, -4)\) and \((6, 4)\) lies on the \(x\)-axis.