Finding The Value Of X On Line Segment AC

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In geometry, understanding the relationships between line segments is fundamental. This article delves into a problem involving line segments and algebraic expressions, providing a step-by-step solution and insightful explanations. We will explore how to find the value of x when given the lengths of segments on a line.

Problem Statement

Point B is located on line segment AC‾{\overline{AC}}. We are given the following lengths:

  • AB=x+6{AB = x + 6}
  • BC=x+8{BC = x + 8}
  • AC=10{AC = 10}

Our goal is to determine the value of x.

Understanding the Problem

The key to solving this problem lies in the Segment Addition Postulate. This postulate states that if a point B lies on the line segment AC‾{\overline{AC}}, then the sum of the lengths of the segments AB‾{\overline{AB}} and BC‾{\overline{BC}} is equal to the length of the segment AC‾{\overline{AC}}. In mathematical terms:

AB+BC=AC{AB + BC = AC}

This fundamental concept allows us to set up an equation and solve for the unknown variable x. By visualizing the line segment and understanding the relationship between its parts, we can approach the problem with clarity and precision.

Applying the Segment Addition Postulate

Now, let's apply the Segment Addition Postulate to our problem. We know that:

  • AB=x+6{AB = x + 6}
  • BC=x+8{BC = x + 8}
  • AC=10{AC = 10}

Substituting these values into the equation AB+BC=AC{AB + BC = AC}, we get:

(x+6)+(x+8)=10{(x + 6) + (x + 8) = 10}

This equation forms the basis for our solution. By simplifying and solving for x, we can find the value that satisfies the given conditions. The subsequent steps involve algebraic manipulation to isolate x and arrive at the final answer.

Solving for x

To solve for x, we need to simplify the equation we derived from the Segment Addition Postulate:

(x+6)+(x+8)=10{(x + 6) + (x + 8) = 10}

Step 1: Combine Like Terms

First, combine the x terms and the constant terms on the left side of the equation:

x+x+6+8=10{x + x + 6 + 8 = 10}

2x+14=10{2x + 14 = 10}

Step 2: Isolate the Variable Term

Next, we want to isolate the term with x. To do this, subtract 14 from both sides of the equation:

2x+14−14=10−14{2x + 14 - 14 = 10 - 14}

2x=−4{2x = -4}

Step 3: Solve for x

Finally, divide both sides of the equation by 2 to solve for x:

2x2=−42{\frac{2x}{2} = \frac{-4}{2}}

x=−2{x = -2}

Therefore, the value of x is -2. This solution aligns with the algebraic steps we've taken, ensuring a logical and accurate result. The negative value of x indicates a specific relationship between the lengths of the segments, which we will discuss further in the next section.

Verification and Implications

Now that we have found the value of x, it's crucial to verify our solution and understand its implications within the context of the problem.

Substituting x = -2

Let's substitute x = -2 back into the expressions for the lengths of the segments:

  • AB=x+6=−2+6=4{AB = x + 6 = -2 + 6 = 4}
  • BC=x+8=−2+8=6{BC = x + 8 = -2 + 8 = 6}

Now, let's check if the Segment Addition Postulate holds true:

AB+BC=4+6=10{AB + BC = 4 + 6 = 10}

Since AB+BC=10{AB + BC = 10} and AC=10{AC = 10}, our solution is verified. The lengths of the segments are consistent with the given information, confirming the accuracy of our calculation. This verification step is essential in ensuring the reliability of the solution and demonstrating a thorough understanding of the problem.

Implications of x = -2

The value x = -2 provides valuable insights into the geometry of the problem. While the lengths AB and BC are positive (4 and 6, respectively), the negative value of x indicates how these lengths are expressed algebraically. It's important to note that segment lengths themselves are always positive, but the variable x can take on negative values depending on the specific algebraic relationships defined in the problem.

In this case, x = -2 satisfies the equation derived from the Segment Addition Postulate, ensuring that the sum of the lengths of segments AB‾{\overline{AB}} and BC‾{\overline{BC}} equals the length of segment AC‾{\overline{AC}}. This understanding reinforces the connection between algebra and geometry, highlighting how algebraic solutions can provide meaningful information about geometric figures.

Conclusion

In summary, we successfully found the value of x in the given problem by applying the Segment Addition Postulate and solving the resulting algebraic equation. We determined that x = -2, and we verified this solution by substituting it back into the original expressions and confirming that the segment lengths are consistent with the given information.

Key Takeaways

  • The Segment Addition Postulate is a fundamental concept in geometry that relates the lengths of segments on a line.
  • Algebraic equations can be used to represent and solve geometric problems.
  • Verifying solutions is crucial to ensure accuracy and understanding.
  • The value of a variable in a geometric context can have implications beyond the numerical value itself.

This problem illustrates the power of combining geometric principles with algebraic techniques. By understanding these concepts and practicing problem-solving strategies, one can develop a strong foundation in mathematics and its applications.

Final Answer

The value of x is -2.