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.
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)