Determine whether each set of measures can be the lengths of the sides of a right triangle.

12, 16, 21

The length of the triangle is$12,16,21$

If c, is the longest side of the triangle, a and b are the other two sides of the triangle then:

(i) If $c^2<a^2+b^2$, then the triangle is acute.

(ii) If $c^2=a^2+b^2$, then the triangle is a right triangle.

(iii) If $c^2>a^2+b^2$, then the triangle is obtuse.

The triangle is having sides of measures 12, 16, and 21.

The longest side of the triangle is 21.

Therefore, the value c is 21.

Therefore, the values of a and b are 12 and 16 respectively.

Now, it can be obtained that:

$\begin{array}{c}{a}^2={12}^2=144\\ b^2={16}^2=256\\ c^2={21}^2=441\\ a^2+b^2=144+256=400\end{array}$

It can be noticed that:

$\begin{array}{c}{a}^2+b^2=400\\ a^2+b^2\ne c^2\end{array}$

As, $c^2\ne a^2+b^2$, therefore, the triangle is not a right triangle.

Therefore, the given set of measures is not the lengths of the sides of a right triangle.