find the coordinates of the points of trisection of the line segment AB with A(2,7) and B(-4, -8)
Answers
Step-by-step explanation:
section formula
(x,y)=m1x2+m2x1/m1+m2,m1y2+m2y1/m1+m2
Answer:
The coordinates of the points of trisection of the line segment AB are ( 0, 2 ) and ( - 2, - 3 ).
Step-by-step-explanation:
NOTE: Refer to the attachment for the diagram.
We have given that,
A ( 2, 7 )
B ( - 4, - 8 )
We have to find the coordinates of the points of trisection of the line segment AB.
Let P and Q be the points of trisection of line segment AB.
P ≡ ( x₁ , y₁ )
Q ≡ ( x₂ , y₂ )
Trisection means these points divide the line segment in three equal parts.
∴ AP = PQ = QB - - ( 1 )
Now,
( AP / PB ) = ( AP / PQ + QB ) - - [ P - Q - B ]
⇒ AP / PB = ( AP / AP + AP ) - - [ From ( 1 ) ]
⇒ AP / PB = AP / 2AP
⇒ AP / PB = 1 / 2
⇒ AP : PB = 1 : 2
∴ P divides the line segment AB in the ratio 1 : 2.
Now, by section formula,
x₁ = 1 ( - 4 ) + 2 ( 2 ) / ( 1 + 2 )
⇒x₁ = - 4 + 4 / 3
⇒x₁ = 0 / 3
⇒x₁ = 0
Now,
y₁ = 1 ( - 8 ) + 2 ( 7 ) / ( 1 + 2 )
⇒ y₁ = - 8 + 14 / 3
⇒ y₁ = 6 / 3
⇒ y₁ = 2
∴ The coordinates of P are ( 0, 2 ).
Now,
AQ / QB = ( AP + PQ ) / QB - - [ A - P - Q ]
⇒ AQ / QB = QB + QB / QB - - [ From ( 1 ) ]
⇒ AQ / QB = 2QB / QB
⇒ AQ / QB = 2 / 1
⇒ AQ : QB = 2 : 1
∴ Point Q divides the line segment AB in the ratio 2 : 1.
Now, by section formula,
x₂ = 2 ( - 4 ) + 1 ( 2 ) / ( 2 + 1 )
⇒ x₂ = - 8 + 2 / 3
⇒ x₂ = - 6 / 3
⇒ x₂ = - 2
Now,
y₂ = 2 ( - 8 ) + 1 ( 7 ) / ( 2 + 1 )
⇒ y₂ = - 16 + 7 / 3
⇒ y₂ = - 9 / 3
⇒ y₂ = - 3
∴ The coordinates of point Q are ( - 2, - 3 ).
─────────────────────
Additional Information:
1. Distance Formula:
The formula which is used to find the distance between two points using their coordinates is called distance formula.
- d ( A, B ) = √[ ( x₁ - x₂ )² + ( y₁ - y₂ )² ]
2. Section Formula:
The formula which is used to find the coordinates of a point which divides a line segment in a particular ratio is called section formula.
- x = ( mx₂ + nx₁ ) / ( m + n )
- y = ( my₂ + ny₁ ) / ( m + n )
