points P and Q are on line segment AB on the same side of the midpoint of AB .Point P divides AB in the ratio 2:3 and Q divides AB in the ratio 3:4.If PQ =2 ,then the length of segment AB is ________
I drew a line AB, with A to the left of B
I marked P and Q on the left side of the midpoint.
"Point P divides AB in the ratio 2:3"
---> let AP = 2x and PB = 3x
then AB = 5x
"Q divides AB in the ratio 3:4"
--> let AQ = 3y and QB = 4y
then AB = 7y
clearly 7y = 5x or y = 5x/7 ***
Given:
PQ = 2
PQ = 3y - 2x
2 = 15/7x - 2x
14 = 15x - 14x
x = 14
then AB = 5x = 70
All the other segments can also be found
To find the length of segment AB, we need to determine the ratio of the lengths of AP and PB, as well as the ratio of the lengths of AQ and QB. Let's break down the problem step by step:
1. Given that P divides AB in the ratio 2:3, let's assume the length of AP is 2x and the length of PB is 3x.
2. Similarly, since Q divides AB in the ratio 3:4, we can assume the length of AQ is 3y and the length of QB is 4y.
Now, we are given that PQ = 2. Using this information, we can set up the following equation:
AP + PQ + QB = AB
Substituting the values we assumed earlier:
2x + 2 + 4y = AB
We also know that the length of PQ is 2, so we can rewrite the equation as:
2x + 2 + 4y = 2x + 3x + 2 + 3y + 4y
Simplifying the equation further:
2 + 4y = 5x + 7y
Moving the variables to one side:
5x - 4y = 2
Now, we have a linear equation with two unknowns (x and y). However, we need another equation to solve for both variables.
The information states that points P and Q are on the same side of the midpoint of AB. It implies that the sum of the ratios 2:3 and 3:4 is equal to 1:
2/3 + 3/4 = 1
To simplify this equation, we can find the lowest common denominator of 3 and 4, which is 12:
(8/12) + (9/12) = 12/12
17/12 = 12/12
Now we have the second equation: 17x = 12y
Since we have two equations, we can solve the system of equations simultaneously:
5x - 4y = 2
17x - 12y = 0
By solving these equations, we find that x = 12 and y = 17.
Finally, we can substitute these values back into the equation 2x + 2 + 4y = AB:
2(12) + 2 + 4(17) = AB
24 + 2 + 68 = AB
94 = AB
Therefore, the length of segment AB is 94 units.