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PROVE SECTION FORMULA. 

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

Answers

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 ∆PAQ and ∆BPC
∠PAQ = ∠BPC (pair of corresponding angles)
∠PQA = ∠BCP (90 °)
Hence, ∆PAQ ∼ ∆BPC (AA similarity criterion)
Hope! You got the concept.
Best Wishes @!

Posted by Afzal Husain(student)on 26/12/10

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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 ' .

SECTION FORMULA

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(student)on 26/12/10

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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)]

SECTION FORMULA

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 .

SECTION FORMULA

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(student)on 26/12/10

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