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

10, 12, 15

The length of the triangle is10, 12, 15

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 10, 12, and 15.

The longest side of the triangle is 15.

Therefore, the value c is 15.

Therefore, the values of a and b are 10 and 12 respectively.

Now, it can be obtained that:

$\begin{array}{c}{a}^2={10}^2=100\\ b^2={12}^2=144\\ c^2={15}^2=225\\ a^2+b^2=100+144=244\end{array}$

It can be noticed that:

$\begin{array}{c}{a}^2+b^2=244\\ a^2+b^2\ne c^2\end{array}$

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.