Question

can the sides of a triangle have lengths of 47, 3, and 47? if so, what kind of triangle is it? yes, acute yes, right yes, obtuse no

Explanation:

Step1: Check triangle - inequality theorem

The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
$47 + 3=50>47$, $47 + 47 = 94>3$, $3+47 = 50>47$. So, a triangle can be formed.

Step2: Determine the type of triangle

Since two sides have length 47, it is an isosceles triangle. Let \(a = 3\), \(b = 47\), \(c = 47\). Use the Law of Cosines \(c^{2}=a^{2}+b^{2}-2ab\cos C\). For the angle opposite the side \(a\), \(\cos A=\frac{b^{2}+c^{2}-a^{2}}{2bc}=\frac{47^{2}+47^{2}-3^{2}}{2\times47\times47}=\frac{2\times47^{2}-9}{2\times47^{2}}>0\). Since \(\cos A>0\), angle \(A\) is acute. And because the triangle is isosceles, all angles are acute.

Answer:

yes, acute