Question

can the sides of a triangle have lengths of 21, 47, and 50? 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.
$21 + 47=68>50$, $21 + 50 = 71>47$, $47+50 = 97>21$. So, a triangle can be formed.

Step2: Determine the type of triangle

Let $a = 21$, $b = 47$, $c = 50$. We use the Pythagorean - like inequality. For a triangle with side lengths $a$, $b$, $c$ 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 $c^{2}$:
$a^{2}=21^{2}=441$, $b^{2}=47^{2}=2209$, so $a^{2}+b^{2}=441 + 2209=2650$.
$c^{2}=50^{2}=2500$.
Since $a^{2}+b^{2}>c^{2}$ ($2650>2500$), the triangle is acute.

Answer:

yes, acute