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

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

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

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(student)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(student)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(student)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(student)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(student)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(student)on 26/12/10