# Division of Line Segment

This topic is really easy. This is just a matter of small common sense and your skill to substitute.

**Formula derivation:**

Consider points A(x_{1},y_{1}) and B(x_{2},y_{2}) is divided by point P(x_{p},y_{p}) to a ratio of m:n.

Using the similar triangles we can derive the following formulas:

Note that if *m=n*. The coordinate of *P *is the midpoint of points *A* and *B*.

_{ }

**Sample Problem 1:**

Points A(-1,3) and B(5,7) is bisected by point P. Such that . Find the coordinate of point *P*.

Solution:

We draft the segment on our paper with the points A, P and B. We can solve the coordinate of P using the formula mentioned above.

Solve for x_{p :}

Solve for y_{p} :

The coordinate of *P* is

**Sample Problem 2:**

Find the coordinates of the point which divides the line segment from *(-2,1)* and *(2,3)* in the ratio 3:4.

Solution:

Let the point (-2,1) is A and point (2,3) is B. Since there is no specific location of desired point we have two possible coordinates. The desired points are X and Y since these two points divides the segment AB to 3:4 ratio.

Coordinate of X: Assume in the figure that points Y does not exist.

AX=3 and XB=4

Solve for

Solve for

For coordinate of Y: Assume that X does not exist

Solve for

Solve for

The two points are: (-2/7 , 13/7) and (2/7 , 15/7)

**Sample Problem 3:**

The line segment joining *P _{1}(1,3)* and

*P*is extended through each end by a distance equal to thrice its original length. Find the coordinates of the new endpoints.

_{2}(-2,-4)Solution:

Draft the points P_{1 }and P_{2} and extend both ends such that the extended segment is thrice the length of the original segment.Let B and C are the new endpoints. Let *d* the distance between P1 and P2. Then P1 to C is *3d* the same distance of *3d* from P2 to B.

Solve for coordinates of B: assume that C does not exist

Solve for

, we can cancel *d* here

Solve for

cancel *d*

Solve for coordinates of C: assume that B does not exist

Solve for

cancel *d*

Solve for

cancel *d*

The required answer is *(10,24)* and *(-11,-25)*

### Dan

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May I know the name of the theorem used

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