The given lengths are the legs of the right triangle, so let $a=1$ and $b=\sqrt{3}.$ According to the Pythagorean theorem, we obtain \begin{align*} c^2 &= a^2 + b^2 \\[3pt] c &= \sqrt{a^2 + b^2}. \end{align*}

Substituting in the values for $a$ and $b$ gives us \begin{align} c &= \sqrt{(1)^2 + (\sqrt{3})^2} \\[3pt] &= \sqrt{1 + 3} \\[3pt] &= \sqrt{4} \\[3pt] &= 2. \end{align}

The given lengths are the legs of the right triangle, so let $a=3\,\textrm{cm}$ and $b=4\,\textrm{cm}.$ According to the Pythagorean theorem, we obtain \begin{align*} c^2 &= a^2 + b^2 \\[3pt] c &= \sqrt{a^2 + b^2}. \end{align*}

Substituting in the values for $a$ and $b$ gives us \begin{align*} c &= \sqrt{(3)^2 + (4)^2} \\[3pt] &= \sqrt{9 + 16} \\[3pt] &= \sqrt{25} \\[3pt] &= 5\,\textrm{cm}. \end{align*}

Let $a=24,$ $b=10,$ and $c$ be the length of the hypotenuse. According to the Pythagorean theorem, we obtain \begin{align*} c^2 &= a^2 + b^2 \\[3pt] c &= \sqrt{a^2 + b^2}. \end{align*}

Substituting in the values for $a$ and $b$ gives us \begin{align} c &= \sqrt{(24)^2 + (10)^2} \\[3pt] &= \sqrt{576 + 100} \\[3pt] &= \sqrt{676} \\[3pt] &= 26. \end{align}

Let $a=7,$ $b=24,$ and $c$ be the length of the hypotenuse. According to the Pythagorean theorem, we obtain \begin{align*} c^2 &= a^2 + b^2 \\[3pt] c &= \sqrt{a^2 + b^2}. \end{align*}

Substituting in the values for $a$ and $b$ gives us \begin{align} c &= \sqrt{(7)^2 + (24)^2} \\[3pt] &= \sqrt{49 + 576} \\[3pt] &= \sqrt{625} \\[3pt] &= 25. \end{align}

Here, we know the lengths of the hypotenuse and one leg, so let $b=4$ and $c=9.$ We need to find $a,$ the length of the other leg.

According to the Pythagorean theorem, we obtain \begin{align} c^2 &= a^2 + b^2 \\[3pt] a^2 &= c^2 - b^2 \\[3pt] a &= \sqrt{c^2 - b^2}. \end{align}

Substituting in the values for $b$ and $c$ gives us \begin{align} a &= \sqrt{(9)^2 - (4)^2} \\[3pt] &= \sqrt{81 - 16} \\[3pt] &= \sqrt{65}. \end{align}

Here, we know the lengths of the hypotenuse and one leg, so let $b=8$ and $c=17.$ We need to find $a,$ the length of the other leg.

According to the Pythagorean theorem, we obtain \begin{align} c^2 &= a^2 + b^2 \\[3pt] a^2 &= c^2 - b^2 \\[3pt] a &= \sqrt{c^2 - b^2}. \end{align}

Substituting in the values for $b$ and $c$ gives us \begin{align} a &= \sqrt{(17)^2 - (8)^2} \\[3pt] &= \sqrt{289 - 64} \\[3pt] &= \sqrt{225} \\[3pt] &=15. \end{align}