dterm Question 68, 9.1.35 HW Score: 59.46\%, 44 of 74 points Points: 0 of 1 Save The lengths of the sides of a triangle are \( \sqrt{86}, \sqrt{5} \), and 9 . Classify the triangle as "acute", "right", or "obtuse". Choose the correct type of triangle below. right acute obtuse

To classify the triangle based on the lengths of its sides \( a = \sqrt{86} \), \( b = \sqrt{5} \), and \( c = 9 \) (where \( c \) is the longest side), we can use the triangle inequality and the Pythagorean theorem. First, let's check if these sides can form a triangle: 1. \( \sqrt{86} + \sqrt{5} > 9 \) 2. \( \sqrt{86} + 9 > \sqrt{5} \) 3. \( \sqrt{5} + 9 > \sqrt{86} \) All inequalities hold true, so these sides do form a triangle. Now, we classify it using the relationship \( c^2 \) (the square of the longest side) compared to \( a^2 + b^2 \): - \( c^2 = 9^2 = 81 \) - \( a^2 = (\sqrt{86})^2 = 86 \) - \( b^2 = (\sqrt{5})^2 = 5 \) - \( a^2 + b^2 = 86 + 5 = 91 \) Now, we compare: - Since \( a^2 + b^2 (91) > c^2 (81) \), the triangle is classified as **acute**. So, the correct answer is **acute**. If you want to impress your friends with a math fact, let them know that the classification of triangles isn't just about angles! The famous triangle inequality theorem ensures that the sum of any two sides must be greater than the third side—it's a boundary for forming valid triangles! Next time you're out on a hike, keep an eye out for the triangular shapes in nature. From mountains to the shapes of trees, they often embody triangles—acute, right, or obtuse! You may find that the angles in these structures create beautiful patterns, reminding you of the math behind them. Who knew nature was a geometry whiz?