Ratio Formula

Ratio formula:Let A and B be two given points. Let P be a point on the line segment AB or AB produced. The P divides the line segment AB in to two segments AP and PB. the lengths of AP and PB are AP and PB. These lengths are in some ratio m:n or that is AP:PB =m:n or AP/PB =m/n. If P lies inside AB, we say that P divides AB internally in the ratio m:n. If P lies outside AB, that is P lies on AB produced, then we say P divides AB externally in the ratio m:n.

With a given ratio m:n, the line AB can be divided either internally or externally


Example

Divide the line segment Ab of length 16 units in the ratio 3:5

Solution (Internally)

Let C be the point inside AB such that AC/CB =3/5 (ratio formula) Since the numerator is smaller than the denominator, C is closer to A than B. Then,


We have 5AC= 3BC

5AC = 3(AB-AC)

5AC = 3AB - 3AC

5AC+3AC = 3AB

8AC = 3AB

Here , we know that AB= 16 units. So plug AB = 16 in the above equation

8AC = 3(16)

8AC = 48

Ac = 48/8

AC = 6 units

then CB = AB-AC

CB = 16 - 6

CB = 10 units

Hence C lies inside AB, 6 units from A and 10 units from B. The point C is unique and it divides AB internally in the ratio 3:5


Solution: (Externally)


Let D be the point outside AB such that AD/DB = 3/5 . Since the numerator is smaller than the denominator, D is closer to A than to B.

Now, we have 5AD = 3DB

5AD = 3(AD+AB)

5AD = 3AD + 3AB

5AD - 3AD = 3AB

2AD = 3AB.

we already know that AB=16. SO plug AB = 16 in the above equation.

2AD = 3(16)

2AD =48

AD =48/2

AD= 24 units.

and , DB = DA + AB = 24+16

DB = 40

Hence, D lies out side AB such that 24 units from A and 40 units from B . The point D unique and it divides AB externally in the given ratio 3:5.


Now, it is very clear from the above example that we can use the ratio formula and divide the given line segment in a particular ratio.



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