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Question

Find the coordinates of points which trisect the line segment joining (1,2) and (3,4).

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Solution

Using the section formula, if a point (x,y) divides the line joining the points (x1,y1) and (x2,y2) in the ratio m:n, then

(x,y)=(mx2+nx1m+n,my2+ny1m+n)

Let A (1,2) and B(3,4) be the given points.

Let the points of trisection be P(a,b) and Q(c,d).

Then, AP=PQ=QB=λ(say)

.PB=PQ+QB=2λ

AQ=AP+PQ=2λ

AP:PB=λ:2λ

=1:2

andAQ:QB=2λ:λ
b=2:1

So, P divides AB internally in the ratio 1:2 while Q divides internally in the ratio 2:1 . Thus, the coordinates of P(a,b) and Q(c,d) are

P(a,b)=(1×3+2×11+2,1×4+2×21+2)

P(a,b)=(13,0)

And,
Q(c,d)=(2×3+1×12+1,2×4+1×(2)2+1)

Q(c,d)=(53,2)

Hence, the two points of trisection are (13,0) and (53,2)

REMARK: As Q is the midpoint of BP. So, the coordinates of Q can also be obtained by using the midpoint formula.

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