The triangle has an angle of 90°, therefore it is a right triangle.
In this type of triangle, we can use the Pythagoras' theorem to find a relation between the smaller sides (legs) and the bigger side (hypotenuse).
If the result is false, the triangle is not a right triangle.
So testing each option, we have:
A)
[tex]\begin{gathered} 5,10,10.2\colon \\ 10.2^2=10^2+5^2 \\ 104.04=100+25 \\ 104.04=125\text{ (F)} \end{gathered}[/tex]B)
[tex]\begin{gathered} 5,\frac{5}{3},10\colon \\ 10^2=5^2+(\frac{5}{3})^2 \\ 100=25+\frac{25}{9} \\ 100=27.778\text{ (F)} \end{gathered}[/tex]C)
[tex]\begin{gathered} 5,10,10.3\colon \\ 10.3^2=10^2+5^2 \\ 106.09=100+25 \\ 106.09=125\text{ (F)} \end{gathered}[/tex]D)
[tex]\begin{gathered} 5,5,2.1\colon \\ 5^2=5^2+2.1^2 \\ 25=25+4.41 \\ 25=29.41\text{ (F)} \end{gathered}[/tex]None of the options can represent a 30-60-90 triangle, unless the correct set of values for option B is: {5, 5√3, 10}. Then we would have:
[tex]\begin{gathered} 5,5\sqrt[]{3},10\colon \\ 10^2=(5\sqrt[]{3})^2+5^2 \\ 100=25\cdot3+25 \\ 100=75+25 \\ 100=100\text{ (T)} \end{gathered}[/tex]So, in this case, the option would be B.
