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Q.




What are the coordinates of the points of trisection of the line segment joining (4, –1) and (–2, –3)?

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a

2, −7 3   and 1, −7 3  

b

2, −5 3   and 0, −7 3  

c

2, −5 6   and 0, −7 6  

d

2, −10 3   and 0, −4 3   

answer is B.

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Detailed Solution

Given a line segment joining the points line segment joining (4, –1) and (–2, –3).
We know that points of trisection means that the line is divided in 3 equal lengths.
Let A(4,-1) and B(-2,-3) be divided by C and D in 3 equal parts.
Question ImageWe know that if a point P(x, y) divides a line segment joining the points A( x 1 , y 1 ) and B( x 2 , y 2 )   in the ratio m:n, then the coordinates of P can be calculated as,
P(x, y)= [ m x 2 +n x 1 m+n , m y 2 +n y 1 m+n ]  .
To find coordinates of C, we use the section formula,
AC = 1 3  AB.
AC:CB = 1:2.
Here m=1 and n=2
x 1 =4, y 1 =−1, x 2 =−2, y 2 =−3  
Putting these values in section formula,
C = [ m x 2 +n x 1 m+n , m y 2 +n y 1 m+n ]  ,
we get
C = [ 1(−2)+2(4) 1+2 , 1(−3)+2(−1) 1+2 ]  
C =  2, −5 3  
Now C will be the midpoint of segment AD.
We know that midpoint formula is given by,
M(x, y) = [ x 1 + x 2 2 , y 1 + y 2 2 ]  .
Let point D be (x, y).
Here,
x 1 =4, y 1 =−1, x 2 =x, y 2 =y  .
Using midpoint formula,
C =  4+x 2 , y−1 2  
On comparing their x and y coordinates,
4+x 2 =2, y−1 2 = −5 3 ⇒4+x=4,3y−3=−10 ⇒x=0,y= −7 3  
So, coordinates of D, are 0, −7 3  .
So, the coordinates of points of trisection are 2, −5 3   and 0, −7 3  .
Therefore, the correct option is 2.

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