The mid-point of a line segment is the point of bisection, which means dividing into two parts of equal length. Suppose we want to divide a line segment into three parts of equal length, we have to locate points suitably to effect a trisection of the segment.

**Points of Trisection of a Line Segment**

The mid-point
of a line segment is the point of bisection, which means dividing into two parts
of equal length. Suppose we want to divide a line segment into three parts of equal
length, we have to locate points suitably to effect a trisection of the segment.

For a given
line segment, there are two points of trisection. The method of obtaining this is
similar to that of what we did in the case of locating the point of bisection (i.e.,
the mid-point). Observe the given Fig. 5.31. Here *P* and *Q* are the
points of trisection of the line segment *AB* where *A* is (
*x*_{1} , *y*_{1} ) and *B* is (
*x*_{2} , *y*_{2} ) . Clearly we know that, P is the mid-point
of AQ and Q is the mid-point of PB. Now consider the Δ*ACQ* and Δ*PDB* (Also, can be verified using similarity
property of triangles which will be dealt in detail in higher classes).

*A*′*P*′*
*=*
P*′*Q*′*
*=*
Q*′*B*′

Note that when we divide the segment
into 3 equal parts, we are also dividing the horizontal and vertical legs into three
equal parts.

**Example 5.16**

Find the
points of trisection of the line segment joining (−2,
−1)
and (4, 8) .

*Solution *

Let *A*
(−2,
−1)
and *B* (4, 8) are the given points.

Let P (a,b)
and Q (c, d) be the points of trisection of AB, so that AP=PQ=QB.

By the formula
proved above,

P is the point

**Progress Check**

(i) Find the coordinates of the points of trisection of the line
segment joining ( 4, −1) and ( −2, −3) .

(ii) Find the coordinates of points of trisection of the line segment
joining the point ( 6, −9) and the origin.

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9th Maths : UNIT 5 : Coordinate Geometry : Points of Trisection of a Line Segment | Formula, Steps, Example Solved Problems | Coordinate Geometry | Maths

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