Points P and Q are both in the line segment AB and on the same side of its midpoint. P divides AB in the ratio 2 : 3, and Q divides AB in the ratio 3 : 4. If PQ = 2, then find the length of the line segment AB.
P divides of the line AB=2:3
Q divides of line AB=3:4
The distance between P and Q=2 cm
we have to find the length of AB=?
AB=AP+PB=2x+3x=5x
Or
AB=AQ+QB=3x+4x=7x
LCM of 5x and 7x=35x=AB
AP+PQ=AQ
2x+2=3x
x=2,
The length of AB=35×2 cm=70 cm
To find the length of line segment AB, we need to determine the distances between points A and P, and between points P and Q, and then add them together.
Let's start by assuming the length of line segment AB is x.
Since P divides AB in the ratio 2:3, we can calculate the distance AP as follows: AP = (2/5) * x.
Similarly, Q divides AB in the ratio 3:4, so the distance PQ can be found using the equation: PQ = (3/7) * x.
According to the given information, PQ = 2. By substituting this value into the equation, we get:
2 = (3/7) * x.
To solve for x, we can cross-multiply and then divide both sides of the equation by (3/7) to isolate x:
2 * (7/3) = x.
The left side of the equation simplifies to (14/3), so x = 14/3.
Therefore, the length of line segment AB is 14/3 units.
Answer
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Label the points from left to right: A,P,Q,M,B
AP = 2x
PB = 3x
AQ = 3y
QB = 4y
PQ = 2
AP+PQ=AQ, so AQ-AP=PQ: 3y-2x = 2
PQ+QB=PB, so PB-QB=PQ: 3x-4y = 2
Solving that, we get y=2, so
AB = 7y = 14