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

Step 1: Understanding the Concept:
The mid-point of a line segment joining \((x_{1}, y_{1})\) and \((x_{2}, y_{2})\) is given by the average of the coordinates.
- A point lies on the \(x\)-axis if its \(y\)-coordinate is 0.
- A point lies on the \(y\)-axis if its \(x\)-coordinate is 0.
Step 2: Key Formula or Approach:
Midpoint \(M = \left( \frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2} \right) \)
Step 3: Detailed Explanation:
Let the points be \(A(5, -4)\) and \(B(6, 4)\).
\[ x_{M} = \frac{5 + 6}{2} = \frac{11}{2} = 5.5 \]
\[ y_{M} = \frac{-4 + 4}{2} = \frac{0}{2} = 0 \]
The coordinates of the mid-point are \((5.5, 0)\).
Since the \(y\)-coordinate is 0, this point lies on the \(x\)-axis.
Step 4: Final Answer:
The mid-point lies on the \(x\)-axis.