Question

determine whether each set of measures can be the measures of the sides of a triangle. if so, classify the triangle as acute, obtuse, or right. 30, 40, 50 no yes; acute yes; obtuse yes; right

Explanation:

Step1: Check Triangle Inequality

First, check if the three lengths (30, 40, 50) can form a triangle. For a triangle with side lengths \(a\), \(b\), \(c\) (where \(c\) is the longest side), the sum of any two sides must be greater than the third side. Here, \(30 + 40 = 70>50\), \(30 + 50 = 80>40\), \(40 + 50 = 90>30\), so it can form a triangle.

Step2: Classify the Triangle (Using Pythagorean Theorem)

Let \(a = 30\), \(b = 40\), \(c = 50\) (longest side). Calculate \(a^{2}+b^{2}\) and \(c^{2}\):
\(a^{2}=30^{2} = 900\), \(b^{2}=40^{2}=1600\), so \(a^{2}+b^{2}=900 + 1600=2500\).
\(c^{2}=50^{2}=2500\).
Since \(a^{2}+b^{2}=c^{2}\), by the Pythagorean theorem, the triangle is a right triangle.

Answer:

yes; right