Math Problem Statement
Point C is on line segment start overline, B, D, end overline BD . Given C, D, equals, 2, x, commaCD=2x, B, D, equals, 3, x, plus, 4, commaBD=3x+4, and B, C, equals, 2, x, minus, 1, commaBC=2x−1, determine the numerical length of start overline, B, D, end overline, . BD .
Solution
We are given three relationships between the line segments on line segment :
Since point is on the line segment , the sum of the segments and should equal the total length of . Therefore, we can write the equation:
Substituting the given expressions for , , and :
Now, simplify the left-hand side:
Thus, the equation becomes:
Now, solve for :
Now that we have the value of , we can substitute it into the expression for :
Thus, the length of is .
Would you like more details or have any further questions? Here are five related questions for deeper understanding:
- How would the problem change if ?
- What is the length of when ?
- How can we verify that holds for any ?
- What happens if point is the midpoint of ?
- Can you find the value of when ?
Tip: When solving for unknowns, ensure that all parts of the equation correspond to the correct geometric interpretation.
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Math Problem Analysis
Mathematical Concepts
Algebra
Linear Equations
Geometry
Segments on a Line
Formulas
BC + CD = BD
Solving linear equations
Theorems
Segment Addition Postulate
Suitable Grade Level
Grades 6-8