A Designer Created A Kitchen Floor Tile In A Triangular Shape. Two Sides Of The Tile Have Lengths Of 8 Inches And 9 Inches. What Is The Range Of The Possible Lengths, $x$, Of The Third Side Of The Tile?Select The Correct Answer:A. 8 In <

by ADMIN 238 views

Introduction

In geometry, the triangle inequality theorem is a fundamental concept that helps us determine the possible lengths of the sides of a triangle. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. In this article, we will apply the triangle inequality theorem to find the range of possible lengths of the third side of a triangle with two given sides of lengths 8 inches and 9 inches.

The Triangle Inequality Theorem

The triangle inequality theorem is a mathematical statement that describes the relationship between the lengths of the sides of a triangle. It states that for any triangle with sides of lengths a, b, and c, the following inequalities must hold:

  • a + b > c
  • a + c > b
  • b + c > a

These inequalities ensure that the sum of the lengths of any two sides of a triangle is greater than the length of the third side.

Applying the Triangle Inequality Theorem to the Given Problem

In this problem, we are given two sides of the triangle with lengths 8 inches and 9 inches. We need to find the range of possible lengths of the third side of the triangle. Let's denote the length of the third side as x.

Using the triangle inequality theorem, we can write the following inequalities:

  • 8 + 9 > x
  • 8 + x > 9
  • 9 + x > 8

Simplifying these inequalities, we get:

  • 17 > x
  • x > 1
  • x > -1

Finding the Range of Possible Lengths of the Third Side

From the inequalities above, we can see that the length of the third side x must be greater than 1 and less than 17. This means that the range of possible lengths of the third side is (1, 17).

However, we need to consider the fact that the length of the third side cannot be equal to the sum of the lengths of the other two sides. Therefore, the upper limit of the range is 17 - ε, where ε is a small positive value.

Conclusion

In conclusion, the range of possible lengths of the third side of a triangle with two given sides of lengths 8 inches and 9 inches is (1, 17 - ε). This range is determined by applying the triangle inequality theorem, which ensures that the sum of the lengths of any two sides of a triangle is greater than the length of the third side.

The Final Answer

The final answer is: (1, 17 - ε)

Discussion

The triangle inequality theorem is a fundamental concept in geometry that helps us determine the possible lengths of the sides of a triangle. In this article, we applied the triangle inequality theorem to find the range of possible lengths of the third side of a triangle with two given sides of lengths 8 inches and 9 inches. The range of possible lengths of the third side is (1, 17 - ε), where ε is a small positive value.

Related Topics

  • Triangle inequality theorem
  • Geometry
  • Mathematics
  • Triangle properties

References

  • [1] "Triangle Inequality Theorem" by Math Open Reference
  • [2] "Geometry" by Khan Academy
  • [3] "Triangle Properties" by Math Is Fun