Divide a Line Segment in a Given Ratio

Mathematics – VII

About Lesson

Divide a line segment internally in a given ratio.

Divide a line segment of length $10 c m$ internally in the ratio $3 : 2 .$

Solution

We follow the following step of construction.

Steps of construction
$S t e p I$
Drawn a line segment

A B = 10 c m

by using a ruler.

$S t e p I I$
Drawn any ray making an acute angle

∠ B A X

with

A B

$S t e p I I I$
Along

A X

, mark-off

5= (3 + 2)

points

A_{1}, A_{2}, A_{3}, A_{4}

and

A_{5}

such that

A A_{1} = A_{1} A_{2} = A_{3} A_{4} = A_{4} A_{5}

$S t e p I v$

Join

B A_{5}

$S t e p V$
Through

A_{3}

draw a line

A_{3}

P

parallel to

A_{5}

B

by making an angle equal to

∠ A A_{5} B

A_{3}

intersecting

A B

at a point

P

The point

P

so obtained is the required point, which divides

A B

internally in the ration

3 : 2

Alternative Method for Division of a Line Segment Internally in a given Ratio

We may use the following steps to divide a given line segment

A B

internally in a given ratio

m : n

, where

m

and

n

are natural numbers.

Steps of construction

$S t e p I$
Draw line segment

A B

of given length.

$S t e p I I$
Draw any ray

A X

making an acute angle

∠ B A X

with

A B

$S t e p I I I$
Draw a ray

B Y

, on opposite side of

A X

, parallel to

A X

by making an angle

∠ B A Y

equal to

∠ B A X

Step IV

Mark off

m

points

A_{1}, A_{2},, A_{m}

, on

A X

and

n

points

B_{1}, B_{2},, B_{n}

B Y

such that

A A_{1} = A_{1} A_{2} = . = A_{m - 1} A_{m}

= B B_{1} = B_{1} B_{2} = . = B n - 1 B_{n}

$S t e p V$
Join

A_{m} B_{n}

. Suppose it intersects

A B

P

The point

P

is the required point dividing

A B

in the ratio

m : n

In triangles

A A_{m} P

and

B B_{n} P

, we have

∠ A_{m} A P = ∠ P B B_{n}

and,

A P A_{m} = ∠ B P B_{n}

So, by

A A

similarly criterion, we have

△ A A_{m} P - △ B B_{n} P

\Rightarrow A A ^{m} = A P/BP

\Rightarrow A P = m/n

Divide a line segment internally in a given ratio.

Divide a line segment of length 10 cm internally in the ratio 3 : 2.

Alternative Method for Division of a Line Segment Internally in a given Ratio

Divide a line segment of length $10 c m$ internally in the ratio $3 : 2 .$