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Let A1,A2,A3,...,An are n Points in a pl...

Let `A_1,A_2,A_3,...,A_n` are n Points in a plane whose coordinates are `(x_1,y_1),(x_2,y_2),....,(x_n,y_n)` respectively. `A_1A_2` is bisected at the point `P_1,P_1A_3` is divided in the ratio `1:2` at `P_2,P_2A_4` is divided in the ratio `1:3` at `P_3,P_3A_5` is divided in the ratio `1:4` at `P_4` and the so on until all n points are exhausted. find the coordinates of the final point so obtained.

Text Solution

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To find the coordinates of the final point obtained after successively dividing the segments formed by the points \( A_1, A_2, A_3, \ldots, A_n \), we will follow a systematic approach. ### Step-by-Step Solution: 1. **Find the coordinates of \( P_1 \)**: - The point \( P_1 \) bisects the segment \( A_1A_2 \). - The coordinates of \( P_1 \) can be calculated using the midpoint formula: \[ ...
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Given that A_1,A_2,A_3, A_n are n points in a plane whose coordinates are x_1,y_1),(x_2,y_2) ,(x_n ,y_n), respectively. A_1A_2 is bisected at the point P_1,P_1A_3 is divided in the ratio A :2 at P_2,P_2A_4 is divided in the ratio 1:3 at P_3,P_3A_5 is divided in the ratio 1:4 at P_4 , and so on until all n points are exhausted. Find the final point so obtained.

If A_(1),A_(2),A_(3),…,A_(n) are n points in a plane whose coordinates are (x_(1),y_(1)),(x_(2),y_(2)),(x_(3),y_(3)),…,(x_(n),y_(n)) respectively. A_(1)A_(2) is bisected in the point G_(1) : G_(1)A_(3) is divided at G_(2) in the ratio 1 : 2, G_(3)A_(5) at G_(4) in the1 : 4 and so on untill all the points are exhausted. Show that the coordinates of the final point so obtained are (x_(1)+x_(2)+.....+ x_(n))/(n) and (y_(1)+y_(2)+.....+ y_(n))/(n)

Knowledge Check

  • P is a point on the line segment joining the points (3,2,-1) and (6,2,-2) . If the x-coordinate of P is 5 , then its y-coordinate is

    A
    `1`
    B
    `-1`
    C
    `2`
    D
    `-2`

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