Write proof for the section formula.

A section formula can be either applied internally or externally.

Answer: Section formula for division of coordinates (x₂, y₂) and (x₁, y₁) in the ratio m : n, will be M(x, y) = (mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n).

Let us go through the step-by-step explanation to understand the proof in detail.

Explanation:

Derivation of the Formula

Let A (x₁, y₁) and B (x₂, y₂) be the endpoints of the given line segment AB and C(x, y) be the point that divides AB (internally) in the ratio m:n.

Then, AC / CB = m / n

We want to find the coordinates (x, y) of C.

Now draw perpendiculars from A, C, and B parallel to Y coordinate joining at P, Q, and R on X-axis respectively.

By seeing the above diagram,

AM = PQ = OQ – OP = (x – x₁)

CN = QR = OR – OQ = (x₂ –  x)

CM = CQ – MQ = (y – y₁)

BN = BR – NR = (y₂ – y)

Clearly, we can see that ∆AMC and ∆CNB are similar and, therefore, their sides are proportional by the AA congruence rule.

AC / CB = AM / CN = CM / BN

Now substituting the values in the above relation

m / n = (x – x₁) / (x₂ -x) = (y – y₁) / (y₂ – y)

m / n = (x – x₁) / (x₂ -x) and m / n = (y – y₁) / (y₂ – y)

Solving the 1st condition,

m(x₂ – x) = n(x – x₁)

(m + n)x = (mx₂ + nx₁)

x = (mx₂ + nx₁) / (m + n)

Solving the 2nd condition,

m(y₂ – y) = n(y – y₁)

(m + n)y = (my₂ + ny₁)

y = (my₂ + ny₁) / (m + n)

Thus, we have seen the proof of the section formula.