Question

Which of these set of lengths are not the side lengths of a right triangle: $(36,77,85),(20,99,101),(27,120,123)$ and $(24,33,42)$?

Answer 1

Step-1: Check whether the set $(36,77,85)$ forms the side of a right triangle:

In a right triangle, side lengths form Pythagorean triplets, i.e., the square of the longest side is equal to the sum of the square of the other two sides.

In $(36,77,85)$, the longest side is $85$

${\left(85\right)}^2=7225$

And ${\left(36\right)}^2+{\left(77\right)}^2=1296+5929$

$\Rightarrow$ ${\left(36\right)}^2+{\left(77\right)}^2=7225$

$\Rightarrow$ ${\left(36\right)}^2+{\left(77\right)}^2={\left(85\right)}^2$

Thus, $(36,77,85)$ can be the side lengths of a right triangle.

Step-2: Check whether the set $(20,99,101)$ form the sides of a right triangle:

In $(20,99,101)$, the longest side is $101$

${\left(101\right)}^2=10201$

And ${\left(20\right)}^2+{\left(99\right)}^2=400+9801$

$\Rightarrow$ ${\left(20\right)}^2+{\left(99\right)}^2=10201$

$\Rightarrow$ ${\left(20\right)}^2+{\left(99\right)}^2={\left(101\right)}^2$

Thus, $(20,99,101)$ can be the side lengths of a right triangle.

Step-3: Check whether the set $(27,120,123)$ form the sides of a right triangle:

In $(27,120,123)$, the longest side is $123$

${\left(123\right)}^2=15129$

And ${\left(27\right)}^2+{\left(120\right)}^2=729+14400$

$\Rightarrow$ ${\left(27\right)}^2+{\left(120\right)}^2=15129$

$\Rightarrow$ ${\left(27\right)}^2+{\left(120\right)}^2={\left(123\right)}^2$

Thus, $(27,120,123)$ can be the side lengths of a right triangle.

Step-4: Check whether the set $(24,33,42)$ form the sides of a right triangle:

In $(24,33,42)$, the longest side is $42$

${\left(42\right)}^2=1764$

And ${\left(24\right)}^2+{\left(33\right)}^2=576+1089$

$\Rightarrow$ ${\left(24\right)}^2+{\left(33\right)}^2=1665$

$\Rightarrow$ ${\left(24\right)}^2+{\left(33\right)}^2\ne {\left(42\right)}^2$

Hence, the set of lengths $(24,33,42)$ are not the side lengths of right triangle: