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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 @!

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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'

If

If

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

For x co-ordinate of R, multiply

Thus, Unknown control sequence 'fracm' .

Similarly, for y and z co-ordinates.

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

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i am satisfied Prakhar answer .

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i think Afzal is write answer.

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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,

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

From figure,

Unknown control sequence 'fracz'

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