# Write proof for the section formula.

A section formula can be either applied internally or externally.

### Answer: Section formula for division of coordinates (x_{2}, y_{2}) and (x_{1}, y_{1}) in the ratio m : n, will be M(x, y) = (mx_{2} + nx_{1})/(m + n), (my_{2} + ny_{1})/(m + n).

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

**Explanation:**

**Derivation of the Formula**

Let A (x_{1}, y_{1}) and B (x_{2}, y_{2}) 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 – x_{1})

CN = QR = OR – OQ = (x_{2} – x)

CM = CQ – MQ = (y – y_{1})

BN = BR – NR = (y_{2 }– 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 – x_{1}) / (x_{2} -x) = (y – y_{1}) / (y_{2} – y)

m / n = (x – x_{1}) / (x_{2} -x) and m / n = (y – y_{1}) / (y_{2} – y)

Solving the 1st condition,

m(x_{2 }– x) = n(x – x_{1})

(m + n)x = (mx_{2} + nx_{1})

x = (mx_{2} + nx_{1}) / (m + n)

Solving the 2nd condition,

m(y_{2} – y) = n(y – y_{1})

(m + n)y = (my_{2} + ny_{1})

y = (my_{2} + ny_{1}) / (m + n)