Coordinate geometry — find distance between two points and section formula

Question

(a) Find the distance between the points $A(3, -4)$ and $B(-2, 8)$ .

(b) Find the coordinates of the point that divides the line segment joining $P(2, -3)$ and $Q(4, 5)$ in the ratio $3:1$ internally.

(CBSE 2024, similar pattern)

Solution — Step by Step

Part (a): Distance Formula

For two points $(x_1, y_1)$ and $(x_2, y_2)$ :

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

d = \sqrt{(-2 - 3)^2 + (8 - (-4))^2} = \sqrt{(-5)^2 + (12)^2} = \sqrt{25 + 144} = \sqrt{169} = \mathbf{13 \text{ units}}

Notice: 5, 12, 13 is a Pythagorean triplet. Recognising these saves calculation time.

Part (b): Section Formula

If a point divides the join of $(x_1, y_1)$ and $(x_2, y_2)$ in ratio $m:n$ internally:

\left(\frac{mx_2 + nx_1}{m + n},\ \frac{my_2 + ny_1}{m + n}\right)

Here $m = 3$ , $n = 1$ , $P(2, -3)$ , $Q(4, 5)$ .

x = \frac{3(4) + 1(2)}{3 + 1} = \frac{12 + 2}{4} = \frac{14}{4} = \frac{7}{2}

y = \frac{3(5) + 1(-3)}{3 + 1} = \frac{15 - 3}{4} = \frac{12}{4} = 3

The required point is $\mathbf{\left(\frac{7}{2},\ 3\right)}$ .

Why This Works

The distance formula is a direct application of the Pythagorean theorem. The horizontal gap $(x_2 - x_1)$ and vertical gap $(y_2 - y_1)$ form the legs of a right triangle, and the distance is the hypotenuse.

The section formula gives a weighted average of the coordinates. The point closer to $Q$ (since $m > n$ ) gets coordinates closer to $Q$ ‘s values. When $m = n$ , it reduces to the midpoint formula: $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$ .

Alternative Method

For the distance, you can avoid the square root step by first checking if the differences form a known Pythagorean triplet (3-4-5, 5-12-13, 8-15-17). Here, $5$ and $12$ immediately give $13$ .

For the midpoint (which is section formula with ratio 1:1), just average the coordinates. CBSE often asks: “Find the midpoint, then show it lies on a given line.” Combine the midpoint formula with the line equation for a quick 2-mark solution.

Common Mistake

In the section formula, students swap the order: they multiply $m$ with $(x_1, y_1)$ instead of $(x_2, y_2)$ . Remember — $m$ goes with the second point and $n$ goes with the first point. The ratio $m:n$ means the point is $m$ parts from the first point and $n$ parts from the second, so it is closer to the second point when $m > n$ .