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 (with \(c\) as the longest side), the Pythagorean inequality for obtuse triangles is \(a^{2}+b^{2}

First, from the triangle inequality: \(9 + 14>c\), so \(c<23\). But since it's obtuse and the longest side, \(c>14\) (because it has to be longer than the other given side to be the longest).

Now, apply the obtuse triangle inequality: \(9^{2}+14^{2}277\). Take the square root of 277: \(\sqrt{277}\approx16.64\).

Step2: Combine conditions

We have two conditions: \(14 < c<23\) (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 \(c>\approx16.64\) and \(c < 23\). The smallest whole number greater than 16.64 and less than 23 is 17. Let's check: For \(c = 17\), \(9^{2}+14^{2}=81 + 196 = 277\), and \(17^{2}=289\). Since \(277<289\), it satisfies the obtuse triangle condition. Also, \(9 + 14=23>17\), so triangle inequality is satisfied.

Answer:

B. 17 inches