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# PROVE SECTION FORMULA.

Asked by Rohan Ghodke(takshila school) , on 16/11/13

Here is the proof of the section formula.

Consider any two points A (x 1 , y 1 ) and B (x 2 , y 2 ) and assume that P (x, y) divides AB internally in the ratio m: n i.e. PA: PB = m: n
Draw AR, PS and BT perpendicular to the x-axis. Draw AQ and PC perpendiculars to PS and BT respectively.

In âˆ†PAQ and âˆ†BPC
∠PAQ = ∠BPC    (pair of corresponding angles)
∠PQA = ∠BCP    (90 °)
Hence, âˆ†PAQ ∼ âˆ†BPC  (AA similarity criterion)

Hope! You got the concept.
Best Wishes @!

Posted by Afzal Husain(D A V Public school) on 26/12/10

Let  P ( x 1 y 1 z 1 )  and  Q ( x 2 y 2 z 2 )  be the two given points. Let R(x,y,z) divide [PQ] in the given ratio  m 1 : m 2  such that

Unknown control sequence 'fracPRRQ ' .

Draw PM, QN and RL perpendiculars to  the XY-plane. Through R, draw a straight line ARB parallel to MLN to meet MP (produced) and NQ in points A and B respectively.

[Since PM, QN, RL are perpendiculars to the XY-plane, are parallel,and as they are cut by the line PRQ, they are coplanar. Points M,L, N lie on a straight line which is the intersection of this plane with XY-plane. Therefore, a line through R parallel to MLN also lies in that plane and hence meets MP (produced) and NQ.]

Clearly,  s  APR and RBQ are similar,

Unknown control sequence 'fracPABQ ' ..............(i)

From figure,

PA = M A M P = L R M P = z z 1  and

BQ = N Q N B = N Q L R = z 2 z .

From (i), we get

Unknown control sequence 'fracz '

( m 1 + m 2 ) z = m 1 z 2 + m 2 z Posted by Prakhar Bindal(DEWAN PUBLIC SCHOOL) on 26/12/10

Unknown control sequence 'fracm ' .

Similarly,  Unknown control sequence 'fracm '  and  Unknown control sequence 'fracm ' .

Hence the co-ordinates of R are

Unknown control sequence 'fracm '

## Remark

If  m 1 : m 2 is positive, then segments PR and RQ have same direction and hence R divides [PQ] internally [shown in  fig 1.5(i)]

If  m 1 : m 2  is negative, then segments PR and RQ have opposite directions and hence R divides [PQ] externally [shown in fig 1.5(ii)]

## Rule to write down the co-ordinates of the point which divides the join of two given points in a given ratio:

Draw any line segment and write down the co-ordinates of the given points P and Q at its extremities. Let R(x,y,z) be the point which divides [PQ] in the ratio  m 1 : m 2 .

For x co-ordinate of R, multiply  m 1 with  x 2 and  m 2  with  x 1  as shown in fig 1.6 by arrow heads and add the products. Divide the sum by  m 1 + m 2 .

Thus,  Unknown control sequence 'fracm ' .

Similarly, for y and z co-ordinates.

## Remark

For problems in which it is required to find out the ratio when a given point divides the join of two given points, it is convenient to take the ratio as  k : 1 ( k = 1) , for, in this way to two unknown ( m 1 and  m 2 ) are reduced to one and the co-ordinates of R become

k +1 k x 2 + x 1 k +1 k y 2 + y 1 k +1

Posted by Prakhar Bindal(DEWAN PUBLIC SCHOOL) on 26/12/10