Pythagoras' Theorem

The hypotenuse is the side opposite the right angle, so we use Pythagoras' theorem $a^2+b^2=c^2$ with $a=3$ and $b=4$.

Calling the hypotenuse x: $3^2+4^2=x^2$, so $9+16=x^2$ and $x^2=25$.

The final step is to square root, because $x^2$ is not the length itself: $x=\sqrt{25}=5$.

The hypotenuse is the side opposite the right angle, so we use Pythagoras' theorem $a^2+b^2=c^2$ with $a=6$ and $b=8$.

Calling the hypotenuse x: $6^2+8^2=x^2$, so $36+64=x^2$ and $x^2=100$.

Square rooting both sides gives the length itself: $x=\sqrt{100}=10$.

The hypotenuse is the side opposite the right angle, so we use Pythagoras' theorem $a^2+b^2=c^2$ with $a=9$ and $b=12$.

Calling the hypotenuse x: $9^2+12^2=x^2$, so $81+144=x^2$ and $x^2=225$.

Square rooting both sides gives the length itself: $x=\sqrt{225}=15$.

The hypotenuse is the side opposite the right angle, and it is always the longest side of a right-angled triangle. Here the longest side is $20$, so $20$ is the hypotenuse.

You can check this with Pythagoras' theorem: $12^2+16^2=144+256=400$, and $20^2=400$, so the two shorter sides are $12$ and $16$.

With $a^2+b^2=c^2$, the hypotenuse $c=13$ and one shorter side is $12$. Because the missing side is a shorter side, we subtract rather than add.

Calling the missing side x: $12^2+x^2=13^2$, so $144+x^2=169$ and $x^2=169-144=25$.

Square rooting gives $x=\sqrt{25}=5$.

Draw a right-angled triangle with the line joining the points as its hypotenuse. The horizontal side is $8-2=6$ and the vertical side is $10-2=8$.

Calling the distance x: $6^2+8^2=x^2$, so $36+64=x^2$ and $x^2=100$.

Square rooting gives $x=\sqrt{100}=10$. The slanted distance is longer than either the horizontal or the vertical gap, but shorter than the two added together.