Question

two sides of an obtuse triangle measure 12 inches and 14 inches. the longest side measures 14 inches. what is the greatest possible whole - number length of the unknown side? 9 inches 3 inches 2 inches 7 inches

Explanation:

Step1: Recall triangle - inequality theorem

For a triangle with side lengths \(a\), \(b\), and \(c\), \(|a - b|\lt c\lt a + b\). Here \(a = 12\) and \(b = 14\), so \(|12-14|\lt c\lt12 + 14\), which simplifies to \(2\lt c\lt26\).

Step2: Use the property of an obtuse - triangle

In an obtuse - triangle, if \(c\) is the longest side, then \(a^{2}+b^{2}\lt c^{2}\). Since 14 is the longest side, let the unknown side be \(x\). Then \(x^{2}+12^{2}\lt14^{2}\), so \(x^{2}+144\lt196\), and \(x^{2}\lt196 - 144=52\), \(x\lt\sqrt{52}\approx7.21\).

Step3: Find the greatest whole - number value

Since \(x\) is a whole number and \(x\lt7.21\), the greatest whole - number value of \(x\) is 7.

Answer:

7 inches