two sides of an obtuse triangle measure 9 inches and 14 inches. the length of longest side is unknown. what…

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), (a + c>b), (b + c>a). For an obtuse triangle, if (c) is the longest side, then (a^{2}+b^{2}<c^{2}) (when the angle opposite (c) is obtuse). Here, two sides are (a = 9), (b = 14), and (c) is the unknown longest side.

Step2: Apply triangle inequality first

From triangle inequality, (9 + 14>c) (since (c) is the longest side, (c>14) and (c>9), so the relevant inequality is (9 + 14>c) i.e., (c<23). But also, for obtuse triangle, (9^{2}+14^{2}<c^{2}). Calculate (9^{2}+14^{2}=81 + 196=277). So (c^{2}>277), then (c>\sqrt{277}\approx16.64). Since (c) is a whole number and (c<23) (from triangle inequality) and (c>\approx16.64), and (c) is the longest side (so (c > 14)). We need the smallest whole number (c) that satisfies (c>\sqrt{277}\approx16.64) and (c<23) and (c) is the longest side.

Step3: Check the options

• Option 16 inches: (16^{2}=256), (9^{2}+14^{2}=277), (256<277) so not obtuse (since (a^{2}+b^{2}>c^{2}) would be acute or right, but we need obtuse so (a^{2}+b^{2}<c^{2})).
• Option 17 inches: (17^{2}=289), (9^{2}+14^{2}=277), (289>277)? Wait no, (277<289) so (9^{2}+14^{2}<17^{2}), and check triangle inequality: (9 + 14=23>17), and (17>14), (17>9) so it is the longest side. Also, (17) is a whole number, and it is the smallest among the options that satisfies (c>\approx16.64) and (c<23) and (9^{2}+14^{2}<c^{2}).

Answer:

17 inches