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How to find the Trisection Points of a line?

  • Last Updated : 22 Dec, 2021

Geometry is a branch of mathematics that deals with lines, angles, segments, points, etc, and helps us to determine the spatial relationship between two different spaces. There are many types of geometry that can be studied. Some of them are Euclidean geometry, Topology, Spherical geometry, Hyperbolic geometry, Differential geometry, and Projective geometry. Let’s learn about line segment,

Line segment

The figure of geometry has two endpoints but no thickness. One can measure the length of the segment but not a line. The endpoints can be named for example AB to determine the segments of different lines.

Properties of the line segment

  • They are not empty set but is connected.
  • The line segment is considered as part of ordered geometry.
  • The length of line segments is fixed according to its figure as it has endpoints.

 Section formula

It is a topic that falls under coordinate geometry. It is used to find the ratio of a line segment divided by a point internally and externally. It is also used in physics to find the center of mass of systems. Mainly we study three formulas under it which are mentioned below:

  1. Internal division
  2. External division
  3. Midpoint formula

How to find the Trisection Points of a line?

Answer:

The formula in which a line is divided into three parts in a certain ratio of 1:2 or 2:1 internally. Use section formula to solve any problem. Section formula is mathematically given by

(\frac{mx_2+nx_1}{m+n},\frac{my_2+ny_1}{m+n})

Where m and n are the two integers of ratio given as m:n.

For the trisection formula, use the section formula twice,

Step 1: Solve the points of the line segment by using the ratio m:n = 1:2.

Step 2: Solve the points of the line segment by using the ratio m:n = 2:1.

Let’s take a look at an example, if the points are given are (3, 2) and (3, 4), according to the trisection rule, the line segment with points (3, 2) and (3, 4) will be divided into the ratios of 1:2 and 2:1.

Now,

(x1, y1) = (3, 2)

(x2, y2) = (3, 4)

For the ratio 1:2

m:n = 1:2

(\frac{mx_2+nx_1}{m+n},\frac{my_2+ny_1}{m+n})

= (1x3 + 2x3/1 + 2, 1x4 + 2x2/1 + 2)

= ((3 + 6)/3 , (4 + 4)/3)

= (3, 8/3)

Then, for ratio 2:1

(\frac{mx_2+nx_1}{m+n},\frac{my_2+ny_1}{m+n})

= (2x3 + 1x3/2 + 1, 2x4 + 1x2/2 + 1)

= ((6 + 3)/3, (8 + 2)/3)

= (3, 10/3)

Sample Question

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

=>(mx2+nx1/m+n , my2+ny1/m+n)

=>(1×3+2×4/1+2, 1×1+2x(-2)/1+2)

=>(3+8/3, 1-4/3)

=>11/3, -1)

Then, for the ratio 2:1

m:n=2:1

=>(mx2+nx1/m+n , my2+ny1/m+n)

=>(2×3+1×4/2+1, 2×1+1x(-2)/2+1)

=>(6+4/3, 2-2/3)

=>(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

=>(mx2+nx1/m+n , my2+ny1/m+n)

=>(1x(-7)+2×5/1+2, 1x(5)+2x(-6)/1+2)

=>(-7+10/3, 5-12/3)

=>(1,-7/2)

For the ratio 2:1

m:n=2:1

=>(mx2+nx1/m+n , my2+ny1/m+n)

=>(2x(-7)+1×5/2+1, 2x(5)+1x(-6)/2+1)

=>(-14+5/3, 10-6/3)

=>(-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

=>(mx2+nx1/m+n , my2+ny1/m+n)

=>(1×1+2×2/1+2, 1x(-8)+2×5/1+2)

=>(1+4/3,-8+10/3)

=>(5/3, 2/3)

For the ratio 2:1

m:n=2:1

=>(mx2+nx1/m+n , my2+ny1/m+n)

=>(2×1+1×2/2+1, 2x(-8)+1×5/2+1)

=>(2+2/3, -16+5/3)

=>(4/3,-11/3)

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