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 the y-coordinates of two points are equal in magnitude but opposite in sign (like $-4$ and $4$), their average is always 0.
This immediately implies that the midpoint will have $y = 0$, which places it directly on the x-axis, saving you calculation time!

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We are given two points, $P(5, -4)$ and $Q(6, 4)$.
We need to find the coordinates of the midpoint of the line segment joining these two points and determine which geometric axis or position it lies on.

Step 2: Key Formula or Approach:
The coordinates of the midpoint $M(x, y)$ of a line segment joining the points $(x_1, y_1)$ and $(x_2, y_2)$ are given by the Midpoint Formula:
\[ M(x, y) = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
We also know that:
- A point lies on the x-axis if its y-coordinate is 0, i.e., of the form $(x, 0)$.
- A point lies on the y-axis if its x-coordinate is 0, i.e., of the form $(0, y)$.
- The origin has coordinates $(0, 0)$.

Step 3: Detailed Explanation:

• Identify the coordinates of the given points:
Let $(x_1, y_1) = (5, -4)$
Let $(x_2, y_2) = (6, 4)$

• Calculate the x-coordinate of the midpoint:
\[ x = \frac{x_1 + x_2}{2} \]
\[ x = \frac{5 + 6}{2} = \frac{11}{2} = 5.5 \]

• Calculate the y-coordinate of the midpoint:
\[ y = \frac{y_1 + y_2}{2} \]
\[ y = \frac{-4 + 4}{2} = \frac{0}{2} = 0 \]

• Write down the complete coordinates of the midpoint:
\[ M(x, y) = \left( 5.5, 0 \right) \]

• Analyze the coordinates of $M$:
Since the y-coordinate of the midpoint is exactly 0, the point must lie on the horizontal coordinate axis, which is the x-axis.


Step 4: Final Answer:
The midpoint of the segment lies on the x-axis.
Hence, option (A) is correct.