Question

can the sides of a triangle have lengths of 34, 21, and 22? if so, what kind of triangle is it? yes, acute yes, right yes, obtuse no

Explanation:

Step1: Check triangle - inequality theorem

The triangle - inequality theorem states that for any triangle with side lengths \(a\), \(b\), and \(c\), \(a + b>c\), \(a + c>b\), and \(b + c>a\).
Let \(a = 21\), \(b = 22\), and \(c = 34\).
\(21+22 = 43>34\), \(21 + 34=55>22\), \(22 + 34 = 56>21\). So, a triangle can be formed.

Step2: Determine the type of triangle

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}Here, \(a = 21\), \(b = 22\), \(c = 34\).
\(a^{2}+b^{2}=21^{2}+22^{2}=441 + 484=925\).
\(c^{2}=34^{2}=1156\).
Since \(925<1156\) (i.e., \(a^{2}+b^{2}

Answer:

C. yes, obtuse