A triangle has side lengths measuring $20 cm, 5 cm$, and $n cm$. Which describes the possible values of $n$?
A. $5 < n < 15$
B. $5 \leq n \leq 15$
C. $15 < n < 25$
D. $15 \leq n \leq 25

Apply the triangle inequality theorem: The sum of any two sides of a triangle must be greater than the third side.
Set up three inequalities based on the triangle inequality theorem using the given side lengths 20, 5, and n.
Solve each inequality to find the constraints on n.
Combine the inequalities to determine the range of possible values for n: 15 < n < 25 ​ .

Explanation

Problem Analysis and Setup We are given a triangle with side lengths 20 cm, 5 cm, and n cm. We need to find the possible values of n. 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. We will apply this theorem to find the possible range for n.

Applying the Triangle Inequality Theorem Applying the triangle inequality theorem:

The sum of 20 cm and 5 cm must be greater than n cm: n"> 20 + 5 > n n"> 25 > n or n < 25

The sum of 20 cm and n cm must be greater than 5 cm: 5"> 20 + n > 5 5 - 20"> n > 5 − 20 -15"> n > − 15 Since n represents a side length, it must be positive, so this inequality is always true.

The sum of 5 cm and n cm must be greater than 20 cm: 20"> 5 + n > 20 20 - 5"> n > 20 − 5 15"> n > 15

Combining the Inequalities Combining the inequalities, we have:


n < 25 and 15"> n > 15
Therefore, the possible values of n are between 15 and 25: 15 < n < 25

Final Answer The possible values of n are described by the inequality 15 < n < 25 .

Examples
The triangle inequality theorem is a fundamental concept in geometry and has practical applications in various fields. For example, in construction, when building a triangular structure, the lengths of the sides must satisfy the triangle inequality to ensure the structure is stable. If the inequality is not satisfied, the structure will not be able to form a triangle and will collapse. Similarly, in navigation, understanding the triangle inequality helps in determining the shortest path between two points, as the sum of any two sides of a triangle is always greater than the third side.

Answered by GinnyAnswer | 2025-07-07

Using the Triangle Inequality Theorem, we found that for the triangle with sides measuring 20 cm, 5 cm, and n cm, the value of n must satisfy the inequalities leading to the conclusion that 15 < n < 25. Therefore, the correct answer is option C: 15 < n < 25.
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Answered by Anonymous | 2025-08-16