A triangle has side lengths measuring [tex]$20 cm, 5 cm$[/tex], and [tex]$n cm$[/tex]. Which describes the possible values of [tex]$n$[/tex]?
A. [tex]$5\ \textless \ n\ \textless \ 15$[/tex]
B. [tex]$5\ \textless \ n\ \textless \ 20$[/tex]
C. [tex]$15\ \textless \ n\ \textless \ 20$[/tex]
D. [tex]$15\ \textless \ n\ \textless \ 25$[/tex]
Answer (1)
Apply the triangle inequality theorem: The sum of any two sides of a triangle must be greater than the third side.
Set up the inequalities: n"> 20 + 5 > n , 5"> 20 + n > 5 , and 20"> 5 + n > 20 .
Solve the inequalities: n < 25 , -15"> n > − 15 , and 15"> n > 15 .
Combine the results to define the range for n : 15 < n < 25 .
Explanation
Problem Analysis Let's analyze the problem. We are given a triangle with two sides of length 20 cm and 5 cm, and we need to find the possible range of values for the third side, n , using the triangle inequality theorem.
Applying the Triangle Inequality Theorem The triangle inequality 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. This gives us three inequalities:
n"> 20 + 5 > n
5"> 20 + n > 5
20"> 5 + n > 20
Solving the Inequalities Let's solve each inequality:
n \implies 25 > n"> 20 + 5 > n ⟹ 25 > n or n < 25
5 \implies n > 5 - 20 \implies n > -15"> 20 + n > 5 ⟹ n > 5 − 20 ⟹ n > − 15 . Since n is a side length, 0"> n > 0 , so this inequality is always true if the other inequalities are satisfied.
20 \implies n > 20 - 5 \implies n > 15"> 5 + n > 20 ⟹ n > 20 − 5 ⟹ n > 15
Combining the Results Combining the inequalities n < 25 and 15"> n > 15 , we find the range of possible values for n :
15 < n < 25
Final Answer Therefore, the possible values of n are described by the inequality 15 < n < 25 .
Examples
The triangle inequality is a fundamental concept in geometry and has practical applications in various fields. For example, in construction, when building a triangular structure like a roof truss, the lengths of the beams must satisfy the triangle inequality to ensure the structure's stability. If the inequality is not satisfied, the structure will collapse. Similarly, in navigation, the shortest distance between two points is a straight line, and any detour will always be longer, illustrating the triangle inequality.
