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: Spot the shortcut before doing any arithmetic.
Look at the $y$-coordinates of the two given points first, before touching the midpoint formula at all.

Step 2: Compare the two y-coordinates.
The points are $(5, -4)$ and $(6, 4)$. Their $y$-coordinates are $-4$ and $4$, and these two numbers are exact negatives of each other. Whenever two numbers are exact negatives of each other, their average is always 0, because $\frac{k + (-k)}{2} = 0$ for any number $k$. So we can already tell, without finishing the formula, that the midpoint's $y$-coordinate will land on 0.

Step 3: Confirm with the midpoint formula.
The midpoint formula for the $y$-coordinate is:
\[ y = \frac{y_1 + y_2}{2} = \frac{-4 + 4}{2} = \frac{0}{2} = 0 \]
This confirms the shortcut. We can also find the $x$-coordinate for completeness:
\[ x = \frac{5 + 6}{2} = \frac{11}{2} = 5.5 \]
So the midpoint is $(5.5, 0)$.

Step 4: Decide which axis this point sits on.
A point sits on the $x$-axis whenever its $y$-coordinate is 0, no matter what its $x$-coordinate is. Since $y = 0$ here, the midpoint lies on the $x$-axis.

Step 5: Final answer.
The midpoint lies on the $x$-axis, which is option (A).
\[ \boxed{x\text{-axis}} \]