More Question on Point of Trisections

Question 1. Find the trisection of the points (4,-2) and (3, 1).

Solution:

According to the trisection rule, the line segment with points (4,-2) and (3,1) will be divided into the ratios of 1:2 and 2:1.

Now,

(x₁, y₁) = (4,-2)

(x₂, y₂) = (3,1)

For the ratio 1:2

m : n = 1 : 2

Coordinate of Points = [(mx₂ + nx₁)/(m+n) , (my₂+ny₁)/(m+n)]

⇒ Coordinate of Points = [(1 × 3+2 × 4)/(1+2), (1 × 1+2 × (-2))/(1+2)]

⇒ Coordinate of Points = [(3+8)/3, (1-4)/3] = (11/3, -1)

Then, for the ratio 2:1

m : n = 2 : 1

Coordinate of Points = [(mx₂ + nx₁)/(m+n) , (my₂+ny₁)/(m+n)]

⇒ Coordinate of Points = [(2 × 3 + 1 × 4)/(2 + 1), (2 × 1 + 1 × (-2))/(2+1)]

⇒ Coordinate of Points = (6+4/3, 2-2/3) = (10/3, 0)

Thus, point of trisections are (11/3, -1) and (10/3, 0).

Question 2. Find the trisection of the points (5, -6) and (-7, 5).

Solution:

According to the trisection rule, the line segment with points (5,-6) and (-7,5) will be divided into the ratios of 1:2 and 2:1.

Now,

(x₁, y₁) = (5,-6)

(x₂, y₂) =(-7,5)

For the ratio 1:2

m:n=1:2

Coordinate of Points = [(mx₂ + nx₁)/(m+n) , (my₂+ny₁)/(m+n)]

⇒ Coordinate of Points = (1 × (-7)+2 × 5/1+2, 1 × (5)+2 × (-6)/1+2)

⇒ Coordinate of Points = (-7+10/3, 5-12/3) = (1,-7/2)

For the ratio 2:1

m:n=2:1

Coordinate of Points = [(mx₂ + nx₁)/(m+n) , (my₂+ny₁)/(m+n)]

⇒ Coordinate of Points = (2 × (-7)+1 × 5/2+1, 2 × (5)+1 × (-6)/2+1)

⇒ Coordinate of Points = (-14+5/3, 10-6/3) = (-3,4/3)

Thus, point of trisections are (1,-7/2) and (-3,4/3).

Question 3. Find the trisection of the points (2, 5) and (1, -8).

Solution:

According to the trisection rule, the line segment with points (2,5) and (1,-8) will be divided into the ratios of 1:2 and 2:1.

Now,

(x₁, y₁) = (2,5)

(x₂, y₂) = (1,-8)

For the ratio 1:2

m:n=1:2

Coordinate of Points = [(mx₂ + nx₁)/(m+n) , (my₂+ny₁)/(m+n)]

⇒ Coordinate of Points = (1 × 1+2 × 2/1+2, 1 × (-8)+2 × 5/1+2)

⇒ Coordinate of Points = (1+4/3,-8+10/3)

⇒ Coordinate of Points = (5/3, 2/3)

For the ratio 2:1

m:n=2:1

Coordinate of Points = [(mx₂ + nx₁)/(m+n) , (my₂+ny₁)/(m+n)]

⇒ Coordinate of Points = (2 × 1+1 × 2/2+1, 2 × (-8)+1 × 5/2+1)

⇒ Coordinate of Points = (2+2/3, -16+5/3) = (4/3,-11/3)

Thus, point of trisections are (5/3, 2/3) and (4/3,-11/3).

To find the trisection points of a line segment, you need to divide the segment into three equal parts. This involves finding the points that divide the segment into three equal lengths. In this article, we will answer “How to find the Trisection Points of a Line?” in detail including section formula.