Question

two sides of an obtuse triangle measure 9 inches and 14 inches. the length of longest side is unknown. what is the smallest possible whole - number length of the unknown side? 16 inches 17 inches 24 inches 25 inches

Explanation:

Step1: Recall triangle inequality and obtuse triangle condition

For a triangle with sides \(a\), \(b\), \(c\) (where \(c\) is the longest side), the triangle inequality states \(a + b>c\). For an obtuse triangle, by the Pythagorean inequality, if \(c\) is the longest side, then \(a^{2}+b^{2}

First, from triangle inequality: \(9 + 14>c\Rightarrow c<23\). But since it's the longest side, \(c>14\) (because 14 is one of the given sides and \(c\) must be longer than the other sides to be the longest).

Now, from the obtuse triangle condition: \(9^{2}+14^{2}277\). Take square root: \(c>\sqrt{277}\approx16.64\).

Step2: Combine conditions

We have two conditions for \(c\): \(14 < c<23\) (from triangle inequality, since \(c\) is the longest side, \(c > 14\) and \(c<9 + 14 = 23\)) and \(c>\sqrt{277}\approx16.64\). Also, \(c\) must be a whole number. So we need the smallest whole number \(c\) such that \(c>16.64\), \(c>14\) (which is already satisfied by \(c>16.64\)) and \(c < 23\). The smallest whole number greater than \(16.64\) and less than \(23\) is \(17\).

Answer:

B. 17 inches