In geometry, it’s common to encounter problems where a point divides a line segment into specific ratios, and you’re required to determine unknown variables based on given lengths. A typical example involves a line segment FH with a point G between F and H, dividing FH into segments FG and GH, each expressed in terms of a variable x.
Problem Statement
Consider a line segment FH with a total length of 18 units. Point G lies between points F and H, dividing FH into two segments: FG and GH. The lengths of FG and GH are given as 4x and 2x, respectively. The task is to determine the value of x.
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Solution Approach
To find the value of x, follow these steps:
- Express the total length of FH in terms of x:FH=FG+GHFH = FG + GHFH=FG+GHGiven that FG=4xFG = 4xFG=4x and GH=2xGH = 2xGH=2x, we can write:FH=4x+2xFH = 4x + 2xFH=4x+2xSimplifying this, we get:FH=6xFH = 6xFH=6x
- Set up the equation using the total length of FH:It’s given that the length of FH is 18 units. Therefore:6x=186x = 186x=18
- Solve for x:Divide both sides of the equation by 6:x=186=3x = \frac{18}{6} = 3x=618=3
Thus, the value of x is 3.
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Verification
To verify, substitute x = 3 back into the expressions for FG and GH:
- FG=4x=4×3=12FG = 4x = 4 \times 3 = 12FG=4x=4×3=12 units
- GH=2x=2×3=6GH = 2x = 2 \times 3 = 6GH=2x=2×3=6 units
Adding these gives:
FG+GH=12+6=18FG + GH = 12 + 6 = 18FG+GH=12+6=18
This matches the given total length of FH, confirming that our solution is correct.
Conclusion
By setting up an equation based on the given segment lengths and solving for x, we determined that the value of x is 3. This method can be applied to similar problems involving line segments divided proportionally by intermediate points.
FAQ
- What is the significance of point G in this problem?
Point G divides the line segment FH into two parts, FG and GH, whose lengths are expressed in terms of the variable x. - How do you set up the equation to solve for x?
Sum the expressions for FG and GH to equal the total length of FH, then solve for x. - Can this method be applied to other similar problems?
Yes, this approach is applicable to any problem where a line segment is divided into parts expressed as multiples of a variable. - Why is it important to verify the solution?
Verification ensures that the calculated value of x satisfies all given conditions of the problem. - What if the total length of FH was different?
The same method applies; adjust the equation to reflect the new total length and solve for x accordingly.
