Answered

A triangle has side lengths measuring $20\text{ cm}$, $5\text{ cm}$, and $n\text{ cm}$. Which describes the possible values of $n$?


A$5 < n < 15$
B$5 < n < 20$
C$15 < n < 20$
D$15 < n < 25$

Verified Answer :

  • Apply the Triangle Inequality Theorem to the sides 20, 5, and n.
  • Establish the inequalities \(20+5 > n\), \(20+n > 5\), and \(5+n > 20\).
  • Simplify these to find the range \(15 < n < 25\).
  • The final answer is \(\boxed{15 < n < 25}\).

Explanation

  1. Analyze the Triangle Inequality Theorem conditions.

The Triangle Inequality Theorem states that for any triangle with sides of lengths \(a\), \(b\), and \(c\), the sum of the lengths of any two sides must be strictly greater than the length of the third side. In this problem, the side lengths are given as \(20\text{ cm}\), \(5\text{ cm}\), and \(n\text{ cm}\). These three sides must satisfy the following three conditions:

  1. \(20 + 5 > n\)
  2. \(20 + n > 5\)
  3. \(5 + n > 20\)
  1. Solve the inequalities to determine the range for n.

We solve each of the three inequalities to find the constraints on \(n\):

  1. From \(20 + 5 > n\), we get \(n < 25\).
  2. From \(20 + n > 5\), we get \(n > -15\). Since \(n\) represents a physical side length, it must be greater than 0, so this condition is automatically satisfied for any valid side length.
  3. From \(5 + n > 20\), we get \(n > 15\).

By combining these results, we find that \(n\) must be greater than 15 and less than 25, which gives the range \(15 < n < 25\).

  1. State the final range for n.

Based on our analysis, the possible values for the side length \(n\) are those that satisfy the inequality \(15 < n < 25\). This matches the last given option. Therefore, the possible values of \(n\) are described by \(15 < n < 25\).
The final answer is \(\boxed{15 < n < 25}\).

Examples

Suppose you are building a triangular brace for a shelf. You have two boards measuring 20 inches and 5 inches. To ensure the board lengths can form a triangle, you need to choose a third board of length \(n\). According to the triangle inequality, your third board must be longer than 15 inches but shorter than 25 inches to physically close the triangle. If you pick a board of 18 inches, the triangle can be formed, but if you pick a board of 10 inches or 30 inches, the triangle will not close.

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