points P and Q are both in the line segment AB and on the same side of its mid point p divides AB in the ratio 2:3 and Q divides in the ratio 3:4. if PQ=2 then find the length line segment AB

AP/AB = 2/3

(AP+2)/AB = 3/4

(AP+2)/AP = 9/8
1 + 2/AP = 9/8
2/AP = 1/8
AP = 16

16/AB = 2/3
18/AB = 3/4

AB = 24

To find the length of the line segment AB, we can use the concept of similar triangles.

Let's assume the length of segment AB is represented as x units.

According to the given information, point P divides AB in the ratio 2:3. This means that the distance from point A to point P is 2/(2+3) times the length of AB, and the distance from point P to point B is 3/(2+3) times the length of AB. Therefore, the lengths of AP and PB are (2/5)x and (3/5)x respectively.

Similarly, point Q divides AB in the ratio 3:4. This means that the distance from point A to point Q is 3/(3+4) times the length of AB, and the distance from point Q to point B is 4/(3+4) times the length of AB. Hence, the lengths of AQ and QB are (3/7)x and (4/7)x respectively.

Now, we are given that PQ has a length of 2 units. Since P and Q are on the same line segment and on the same side of its midpoint, we can calculate the distance between the two points by subtracting their positions along the line. Therefore, we have:

PQ = QB - QP = [(4/7)x] - [(3/7)x] = (1/7)x

Given that PQ is 2 units, we can set up an equation:

(1/7)x = 2

To solve for x, we multiply both sides of the equation by 7:

x = 2 * 7 = 14

Hence, the length of line segment AB is 14 units.