Division of Line Segment
This topic is really easy. This is just a matter of small common sense and your skill to substitute.
Consider points A(x1,y1) and B(x2,y2) is divided by point P(xp,yp) 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.
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 xp :
Solve for yp :
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.
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
For coordinate of Y: Assume that X does not exist
The two points are: (-2/7 , 13/7) and (2/7 , 15/7)
Sample Problem 3:
The line segment joining P1(1,3) and P2(-2,-4) is extended through each end by a distance equal to thrice its original length. Find the coordinates of the new endpoints.
Draft the points P1 and P2 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
, we can cancel d here
Solve for coordinates of C: assume that B does not exist
The required answer is (10,24) and (-11,-25)