Which of the following sets of measures could not be the sides of a right triangle?

$\mathit{A}$$\mathbf{(}\mathbf{12}\mathbf{,}\mathbf{16}\mathbf{,}\mathbf{24}\mathbf{)}$ $\mathit{C}$ $\mathbf{(}\mathbf{24}\mathbf{,}\mathbf{45}\mathbf{,}\mathbf{51}\mathbf{)}$

$\mathit{B}$$\mathbf{(}\mathbf{10}\mathbf{,}\mathbf{24}\mathbf{,}\mathbf{26}\mathbf{)}$ $\mathit{D}$$\mathbf{(}\mathbf{18}\mathbf{,}\mathbf{24}\mathbf{,}\mathbf{30}\mathbf{)}$

Consider a right triangle with perpendicular $a$, base $b$and hypotenuse $c$.

In right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.That is,

$c^2=a^2+b^2$

Note: Hypotenuse is the largest side of a right angle triangle.

Consider the set $(12,16,24)$.

24 is the largest value in the given set.

For a right angle triangle,

$c^2=a^2+b^2$

Therefore, to check whether the given points satisfies the above condition.

That is to check whether ${\left(24\right)}^2={\left(12\right)}^2+{\left(16\right)}^2$is true or not.

If it is true then the given set of measures are sides of a right angle triangle and if not, then they are not the measures of a right angle triangle.

${\left(24\right)}^2=576$

$\begin{array}{c}{\left(12\right)}^2+{\left(16\right)}^2=144+256\\ =400\end{array}$

See that , ${\left(24\right)}^2\ne {\left(12\right)}^2+{\left(16\right)}^2$

Therefore, this set is not the measure of the sides of a right angle triangle. $\dots \left(1\right)$

Now, consider the set $(10,24,26)$.

26 is the largest value in the given set.

For a right angle triangle,

$c^2=a^2+b^2$

Therefore, to check whether the given points satisfies the above condition.

That is to check whether${\left(26\right)}^2={\left(10\right)}^2+{\left(24\right)}^2$is true or not.

If it is true then the given set of measures are sides of a right angle triangle and if not, then they are not the measures of a right angle triangle.

${\left(26\right)}^2=676$

$\begin{array}{c}{\left(10\right)}^2+{\left(24\right)}^2=100+576\\ =676\end{array}$

See that , ${\left(26\right)}^2={\left(10\right)}^2+{\left(24\right)}^2$

Therefore, this set is the measure of the sides of a right angle triangle. $\dots \left(2\right)$

Now, consider the set $(24,45,51)$.

51 is the largest value in the given set.

For a right angle triangle,

$c^2=a^2+b^2$

Therefore, to check whether the given points satisfies the above condition.

That is to check whether ${\left(51\right)}^2={\left(24\right)}^2+{\left(45\right)}^2$is true or not.

If it is true then the given set of measures are sides of a right angle triangle and if not, then they are not the measures of a right angle triangle.

${\left(51\right)}^2=2601$

$\begin{array}{c}{\left(24\right)}^2+{\left(45\right)}^2=576+2025\\ =2601\end{array}$

See that , ${\left(51\right)}^2={\left(24\right)}^2+{\left(45\right)}^2$

Therefore, this set is the measure of the sides of a right angle triangle. $\dots \left(3\right)$

Now, consider the set $(18,24,30)$.

30 is the largest value in the given set.

For a right angle triangle,

$c^2=a^2+b^2$

Therefore, to check whether the given points satisfies the above condition.

That is to check whether ${\left(30\right)}^2={\left(18\right)}^2+{\left(24\right)}^2$is true or not.

If it is true then the given set of measures are sides of a right angle triangle and if not, then they are not the measures of a right angle triangle.

${\left(30\right)}^2=900$

$\begin{array}{r}{\left(18\right)}^2+{\left(24\right)}^2=324+576\\ =900\end{array}$

See that , localid="1647944034283" ${\left(30\right)}^2={\left(18\right)}^2+{\left(24\right)}^2$

Therefore, this set is the measure of the sides of a right angle triangle. localid="1647944039779" $\dots \left(4\right)$

From$\left(1\right),\left(2\right),\left(3\right)\&\left(4\right)$, it is clear that only the measures in the set A could not be the sides of a right angle triangle.