1 markMCQSection A

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

Q: 8. 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

(a) x-axis — mid-point = $(5.5, 0)$; since $y = 0$ and $x \ne 0$, the point lies on the x-axis.

Q: Is $\left(\frac{11}{2}, 0\right)$ the same as the origin?

No. The origin is $(0, 0)$. Here $x = 11/2 = 5.5$ is not zero, so the point is on the x-axis but well away from the origin.

Coordinate Geometry

Mid-point formula

Options

(A)x-axis

(B)y-axis

(C)origin

(D)neither x-axis nor y-axis

Official Answer

(a) x-axis — mid-point = $(5.5, 0)$ ; since $y = 0$ and $x \ne 0$ , the point lies on the x-axis.

mid-point formulacoordinate geometryx-axisy-coordinate zero(11/2, 0)average of coordinates

Marking Scheme

11 mark: correct option (a) x-axis.
2Justification (internal): mid-point = $\left(\frac{11}{2}, 0\right)$ ; y-coordinate 0 with $x \ne 0$ ⇒ point on x-axis.

Hint

Average the coordinates; if the resulting y-coordinate is $0$ (and $x \ne 0$ ), the point lies on the x-axis.

Quick Oral Answer

The mid-point is 5.5 comma 0; since its y-coordinate is zero but x is non-zero, the point lies on the x-axis.

Analysis & Explanation

The mid-point of $(5, -4)$ and $(6, 4)$ comes out to $(5.5, 0)$ , and any point with y-coordinate 0 (and x ≠ 0) lies on the x-axis.

Concept

Mid-point formula: $\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$ .
A point lies on the x-axis exactly when its y-coordinate is 0.

Key steps

Mid-point = $\left(\frac{5+6}{2}, \frac{-4+4}{2}\right) = (5.5, 0)$ .
$y = 0$ and $x = 5.5 \ne 0$ → point lies on the x-axis, option (a).

Common mistakes

Option (b): confuses the condition — y-axis needs x = 0, but here $x \ne 0$ .
Option (c): origin needs BOTH coordinates zero; here only y is zero.
Missing the quick symmetry that −4 and +4 are equal and opposite, so they cancel to give y = 0 instantly.

Real-world

Recognising that mirror-image y-values (−4 and +4) always average to zero is a fast exam trick worth spotting on sight.

Common Mistakes

1Confusing the conditions: thinking $y = 0$ means the point is on the y-axis (it is actually the x-axis).
2Choosing 'origin' because one coordinate is 0, forgetting the origin needs BOTH coordinates to be 0.
3Subtracting instead of adding coordinates in the formula, e.g. computing $(6-5)/2$ or $(4-(-4))/2$ and misreading the result.

Interesting Facts

The mid-point formula is a special case of the section formula with ratio 1:1 — dividing a segment into two equal parts.

René Descartes, who founded coordinate geometry in the 17th century, reportedly conceived the idea of locating points by coordinates while watching a fly move across his ceiling.

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Frequently Asked Questions

How do I decide if a point lies on the x-axis or y-axis?

A point lies on the x-axis when its y-coordinate is 0, and on the y-axis when its x-coordinate is 0. If both are 0, it is the origin.

Why does the y-coordinate become 0 here?

The two y-values are −4 and +4, which are equal and opposite. Their sum is 0, so their average $(-4+4)/2$ is also 0, placing the mid-point on the x-axis.

Is $\left(\frac{11}{2}, 0\right)$ the same as the origin?

No. The origin is $(0, 0)$ . Here $x = 11/2 = 5.5$ is not zero, so the point is on the x-axis but well away from the origin.