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:

You can quickly identify if the midpoint lies on the $x$-axis by looking at the $y$-coordinates of the given points.
Since $-4$ and $4$ are additive inverses, their average is automatically 0, meaning $y = 0$.
This immediately points to the $x$-axis without even calculating the $x$-coordinate.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The topic is Coordinate Geometry.
We are given two points and need to find the coordinates of their midpoint, and then determine on which coordinate axis or position this midpoint lies.

Step 2: Key Formula or Approach:
The coordinates of the midpoint $M(x, y)$ of a line segment joining two points $A(x_1, y_1)$ and $B(x_2, y_2)$ are given by the Midpoint Formula:
\[ M(x, y) = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
To check where a point lies:

• A point lies on the $x$-axis if its $y$-coordinate is 0 ($y = 0$).

• A point lies on the $y$-axis if its $x$-coordinate is 0 ($x = 0$).

• A point lies on the origin if both coordinates are 0 ($x = 0, y = 0$).


Step 3: Detailed Explanation:

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

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

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

• Combine coordinates to write the midpoint:
\[ M(x, y) = (5.5, 0) \]

• Since the $y$-coordinate is equal to 0, this point is located on the $x$-axis.


Step 4: Final Answer:
The midpoint lies on the $x$-axis, which corresponds to option (A).