Points P,Q,R and S divide the line segment joining the points A(1,2) and B(6,7) in 5 equal parts. Find
the coordinates of the points P,Q and R
Answers
Step-by-step explanation:
Given :-
Points P,Q,R and S divide the line segment joining the points A(1,2) and B(6,7) in 5 equal parts.
To find :-
Find the coordinates of the points P,Q and R ?
Solution :-
Given points are A(1,2) and B(6,7)
Points P,Q,R and S divide the line segment joining the points A(1,2) and B(6,7) in 5 equal parts.
AP+PQ+QR+RS+SB = AB
Total parts = 5
AP : PB = 1:4
AQ : QB = 2:3
AR : RB = 3:2
AS : SB = 4:1
I) Finding the coordinates of P :-
Let (x1, y1) = A(1,2) => x1 = 1 and y1 = 2
Let (x2, y2) = B(6,7) => x2 = 6 and y2 = 7
Let m1:m2 =AP:PB = 1:4 => m1 = 1 and m2 = 4
We know that
Section formula
({m1x2+m2x1}/(m1+m2),{m1y2+m2y1}/(m1+m2) )
=> ({(1)(6)+(4)(1)}/(1+4) , {(1)(7)+(4)(2)}/(1+4) )
=> ( (6+4)/5 , (7+8)/5 )
=> (10/5, 15/5)
=> (2,3)
The coordinates of P = (2,3)
II) Finding the coordinates of Q :-
Let (x1, y1) = A(1,2) => x1 = 1 and y1 = 2
Let (x2, y2) = B(6,7) => x2 = 6 and y2 = 7
Let m1:m2 =AQ:QB= 2:3 => m1 = 2 and m2 = 3
We know that
Section formula
({m1x2+m2x1}/(m1+m2),{m1y2+m2y1}/(m1+m2) )
=> ({(2)(6)+(3)(1)}/(2+3) , {(2)(7)+(3)(2)}/(2+3) )
=> ( (12+3)/5 , (14+6)/5 )
=> (15/5, 20/5)
=> (3,4)
The coordinates of Q = (3,4)
III)Finding the coordinates of R :-
Let (x1, y1) = A(1,2) => x1 = 1 and y1 = 2
Let (x2, y2) = B(6,7) => x2 = 6 and y2 = 7
Let m1:m2 =AR:RB = 3:2 => m1 = 3 and m2 = 2
We know that
Section formula
({m1x2+m2x1}/(m1+m2),{m1y2+m2y1}/(m1+m2) )
=> ({(3)(6)+(2)(1)}/(3+2) , {(3)(7)+(2)(2)}/(3+2) )
=> ( (18+2)/5 , (21+4)/5 )
=> (20/5, 25/5)
=> (4,5)
The coordinates of R = (4,5)
IV)Finding the coordinates of S :-
Let (x1, y1) = A(1,2) => x1 = 1 and y1 = 2
Let (x2, y2) = B(6,7) => x2 = 6 and y2 = 7
Let m1:m2 =AS:SB = 4:1 => m1 = 4 and m2 = 1
We know that
Section formula
({m1x2+m2x1}/(m1+m2),{m1y2+m2y1}/(m1+m2) )
=> ({(4)(6)+(1)(1)}/(4+1) , {(4)(7)+(1)(2)}/(4+1) )
=> ( (24+1)/5 , (28+2)/5 )
=> (25/5, 30/5)
=> (5,6)
The coordinates of S = (5,6)
Answer:-
The coordinates of P = (2,3)
The coordinates of Q = (3,4)
The coordinates of R = (4,5)
Used formulae:-
Section formula:-
The coordinates of the points which divides the linesegment joining the points (x1, y1) and (x2, y2) is
({m1x2+m2x1}/(m1+m2),{m1y2+m2y1}/(m1+m2) )
