Math Problem Statement

Two sides of a triangle are 10 unit and 5 unit long. (The triangle is not necessarily a right triangle!) What values are possible for the length of the third side? Enter your answer as an interval.

Solution

To determine the possible values for the length of the third side of a triangle, we apply the triangle inequality theorem, which states:

For any triangle with sides aa, bb, and cc:

a+b>ca + b > c a+c>ba + c > b b+c>ab + c > a

Given a=10a = 10 and b=5b = 5, let cc be the length of the third side. Applying the inequalities:

  1. 10+5>c10 + 5 > c
    c<15c < 15

  2. 10+c>510 + c > 5
    c>5c > -5 (which is always true for positive lengths)

  3. 5+c>105 + c > 10
    c>5c > 5

Combining the valid conditions, we get:

5<c<155 < c < 15

Thus, the possible values for the length of the third side cc lie in the interval:

(5,15)(5, 15)

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Related Questions:

  1. How does the triangle inequality theorem apply to different types of triangles?
  2. What happens if the given sides are equal (e.g., 10 and 10)?
  3. Can a triangle exist if the sum of two sides equals the third side?
  4. How would the interval change if one given side was shorter (e.g., 4 instead of 5)?
  5. What are the possible angles in a triangle with sides 10, 5, and a third side within this range?

Tip:

The sum of the two shortest sides must always be greater than the longest side for a valid triangle!

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Math Problem Analysis

Mathematical Concepts

Geometry
Triangle Inequality Theorem

Formulas

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

Theorems

Triangle Inequality Theorem

Suitable Grade Level

Grades 7-9