Midpoint = ((x₁+x₂)/2, (y₁+y₂)/2). Systematic calculation for Class 10.

Find the Midpoint of (-2, 3) and (4, -1) — Midpoint Formula

Question

Find the midpoint of the line segment joining the points $(-2, 3)$ and $(4, -1)$ .

Solution — Step by Step

The midpoint $M$ of a segment joining $(x_1, y_1)$ and $(x_2, y_2)$ is:

M = \left(\frac{x_1 + x_2}{2},\ \frac{y_1 + y_2}{2}\right)

This is just the average of the x-coordinates and the average of the y-coordinates — nothing more.

Let $(x_1, y_1) = (-2, 3)$ and $(x_2, y_2) = (4, -1)$ .

Labelling first prevents sign errors, which is where most marks are lost in board exams.

x_M = \frac{x_1 + x_2}{2} = \frac{-2 + 4}{2} = \frac{2}{2} = 1

y_M = \frac{y_1 + y_2}{2} = \frac{3 + (-1)}{2} = \frac{2}{2} = 1

The midpoint of the segment is M = (1, 1).

Why This Works

The midpoint formula is really just the arithmetic mean applied to coordinates. If you have two values on a number line, their average gives the point exactly halfway between them. We apply that idea separately to x and y.

Geometrically, you can verify: the distance from $(-2, 3)$ to $(1, 1)$ equals the distance from $(1, 1)$ to $(4, -1)$ . Both come out to $\sqrt{13}$ , confirming $(1, 1)$ sits exactly in the middle.

This question has appeared in NCERT Exercise 7.2 and is a guaranteed 2-marker in CBSE Class 10 boards. The formula is also the foundation for Section Formula questions — get comfortable with this before moving there.

Alternative Method

You can verify the midpoint by checking equal distances using the distance formula.

Distance from $(-2, 3)$ to $(1, 1)$ :

d_1 = \sqrt{(1-(-2))^2 + (1-3)^2} = \sqrt{9 + 4} = \sqrt{13}

Distance from $(1, 1)$ to $(4, -1)$ :

d_2 = \sqrt{(4-1)^2 + (-1-1)^2} = \sqrt{9 + 4} = \sqrt{13}

Since $d_1 = d_2$ , the point $(1, 1)$ divides the segment equally — confirmed as the midpoint. In exams, skip this verification unless explicitly asked.

Common Mistake

Students write $\frac{-2 + 4}{2} = \frac{-6}{2} = -3$ — treating $-2 + 4$ as $-2 \times 4 = -8$ , or misreading it as $-(2+4)$ . The negation applies only to the 2, not to the sum. Always compute the numerator first before dividing: $-2 + 4 = +2$ , then $\frac{2}{2} = 1$ .

If both coordinates of the midpoint come out as whole numbers (like they do here), that’s a good sign your arithmetic is clean. NCERT problems are designed with integer answers — a decimal result mid-calculation usually signals a sign error.

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

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