Question

can the sides of a triangle have lengths of 36, 40, and 49? if so, what kind of triangle is it? yes, acute yes, right yes, obtuse no

Explanation:

Step1: Check triangle - inequality theorem

For three side - lengths \(a\), \(b\), and \(c\) of a triangle, the following three inequalities must hold: \(a + b>c\), \(a + c>b\), and \(b + c>a\).
Let \(a = 36\), \(b = 40\), and \(c = 49\).
\(36+40 = 76>49\), \(36 + 49=85>40\), \(40 + 49 = 89>36\). So, a triangle can be formed.

Step2: Determine the type of triangle

Use the Pythagorean - related rule. For a triangle with side - lengths \(a\), \(b\), and \(c\) where \(c\) is the longest side, if \(a^{2}+b^{2}=c^{2}\), it is a right - triangle; if \(a^{2}+b^{2}>c^{2}\), it is an acute - triangle; if \(a^{2}+b^{2}Calculate \(a^{2}+b^{2}\) and \(c^{2}\): \(a^{2}=36^{2}=1296\), \(b^{2}=40^{2}=1600\), so \(a^{2}+b^{2}=1296 + 1600=2896\). And \(c^{2}=49^{2}=2401\).
Since \(a^{2}+b^{2}=2896>2401 = c^{2}\), the triangle is acute.

Answer:

yes, acute