## A line segment AB of length 10 cm is divided at P internally in the ratio 2:3. Then find the length of AP ​

Question

A line segment AB of length 10 cm is divided at P internally in the ratio 2:3. Then find the length of AP

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4 weeks 2021-08-18T19:41:26+00:00 1 Answer 0 views 0

1. Step-by-step explanation:

We follow the following step of construction.

Steps of construction

Step I

Drawn a line segment AB=10 cm by using a ruler.

Step II

Drawn any ray making an acute angle ∠BAX with AB.

Step III

Along AX, mark-off 5(=3+2) points A

1

,A

2

,A

3

,A

4

and A

5

such that

AA

1

=A

1

A

2

=A

3

A

4

=A

4

A

5

.

Step Iv

Join BA

5

Step v

Through A

3

draw a line A

3

P parallel to A

5

B by making an angle equal to ∠AA

5

B at A

3

intersecting AB at a point P.

The point P so obtained is the required point, which divides AB 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 AB internally in a given ratio m:n, where m and n are natural numbers.

Steps of construction

Step I

Draw line segment AB of given length.

Step II

Draw any ray AX making an acute angle ∠BAX with AB.

Step III

Draw a ray BY, on opposite side of AX, parallel to AX by making an angle ∠BAY equal to ∠BAX.

Step IV

Mark off m points A

1

,A

2

,,A

m

, on AX and n points B

1

,B

2

,,B

n

on BY such that

AA

1

=A

1

A

2

=.=A

m−1

A

m

=BB

1

=B

1

B

2

=.=Bn−1B

n

.

Step V

Join A

m

B

n

. Suppose it intersects AB at P.

The point P is the required point dividing AB in the ratio m:n.

In triangles AA

m

P and BB

n

P, we have

∠A

m

AP=∠PBB

n

and, APA

m

=∠BPB

n

So, by AA similarly criterion, we have

△A A

m

P−△BB

n

P

BB

n

AA

m

=

BP

AP

BP

AP

=

n

m