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Chapter 1: Problem 143

Two sides of a triangle have lengths of \(a\) inches and \(b\) inches, where \(a

Short Answer

Expert verified
The correct expression is \(b-a\). It represents the difference in lengths between the two sides \(b\) and \(a\) with \(b\) being the longer side.

Step by step solution

01

Understand what each expression represents

In this case, \(a-b\) and \(b-a\) are the differences between the lengths of two sides of a triangle. \(a-b\) represents the scenario where we subtract the length of side \(b\) from side \(a\) and \(b-a\) represents the scenario where we subtract the length of side \(a\) from side \(b\).
02

Analyze the given condition

The triangle's condition specifies that \(a<b\), meaning side \(a\) is shorter than side \(b\). If we use the expression \(a-b\), we would end up with a negative result as side \(b\) is longer. Since lengths cannot be negative, this expression does not have a valid meaning in this context.
03

Identify the correct expression

Then comes the \(b-a\) expression. This expression when \(a<b\) results in a positive result which makes sense in the real-world scenario as lengths are always positive. Therefore, \(b-a\) is the correct expression. In the context of the triangle, this represents the difference in lengths between the two sides \(b\) and \(a\) with \(b\) being the longer one.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Difference of Lengths
When working with triangles, one might need to calculate the difference between the lengths of two sides. This is expressed in mathematical terms as either \(a-b\) or \(b-a\). These expressions are straightforward:
  • \(a-b\) means you subtract the length of side \(b\) from side \(a\).
  • \(b-a\) means you subtract the length of side \(a\) from side \(b\).
Understanding these expressions is crucial because they help in comparing the two side lengths. In geometry, especially in triangles, knowing how much longer one side is compared to the other can be very important for solving problems and understanding the properties of the shape.
Positive Result
In the context of triangles, the result of subtracting one side's length from another should always be positive. This is because side lengths cannot be negative in geometry. Hence, a negative result from a subtraction would not make practical sense.

For example, if we have two sides \(a\) and \(b\) with \(a
  • Then \(a-b\) will result in a negative value, which doesn't fit within the rules of geometry.
  • Whereas, \(b-a\) will give a positive value.
    • A positive result is essential since it reflects reality, showing how much one side exceeds the other.
    Valid Meaning in Geometry
    A valid geometric expression is one that aligns with the rules and realities of mathematical principles. In the context of this exercise, a valid or meaningful result should not only obey mathematical laws but also offer insights into the characteristics of the figure, such as understanding the length differences between sides.

    Given that side lengths cannot be negative, the expression selected needs to reflect this. In the exercise, since \(a
    Side Lengths Comparison
    Comparing side lengths is a fundamental task in understanding and applying triangle inequalities. When you compare \(a\) and \(b\) under the condition \(a
  • This gives a direct and quantitative comparison.
  • It helps in calculating necessary measurements like perimeter or checking if a set of lengths can form a triangle.
    • Accurately comparing side lengths allows the application of essential triangle principles, aiding in ensuring robust geometric reasoning and solutions.

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