Groups of three whole numbers that satisfy Pythagoras' theorem are called "Pythagorean triples". Find the third number that would make these into Pythagorean triples. They are given in numerical order.

$8, \_\_\_,17$

Groups of three whole numbers that satisfy Pythagoras' theorem are called "Pythagorean triples". Find the third number that would make these into Pythagorean triples. They are given in numerical order.

$\_\_\_, 40, 41$

You proved Pythagoras' theorem by constructing squares on each of the sides. Does the formula work with other shapes, for example, circles? If semi-circles are constructed on the sides of a right triangle whose sides measure $3\mathrm{cm},4\mathrm{cm}$ and $5\mathrm{cm}$, show that Pythagoras' theorem still applies.

Find the shortest distance between the following pair of points. If the answer to the nearest tenth is $d$ units then find $d$.

$\left(-1,4\right)$ and $(3,10)$

Find the shortest distance between the following pair of points. If the answer to the nearest tenth is $d$ units then find $d$.

$\left(-7,-11\right)$ and $(4,-25)$

Find the shortest distance between the following pair of points. If the answer to the nearest tenth is $d$ units then find $d$.

$\left(0,-6\right)$ and $(4,3)$

Find the shortest distance between the following pair of points. If the answer to the nearest tenth is $d$ units then find $d$.

$\left(5,-2\right)$ and $(13,-2)$

When data is sent electronically, for example when you download a song, you don't want parts of it to be missing. Computer programmers, who imagine data as points in space, use Pythagoras' theorem to make sure that the data is where it should be. Given an original point and its final location, determine whether or not a piece of data is where it  should be (a distanced from its original location). Copy and complete this table. 

In video games, the location of each character is tracked using coordinates. The program uses Pythagoras' theorem to calculate the distance between characters. It then compares this value with minimum distances that have been set for certain actions. For example, a role-playing game is programed to register a "hit" by a character's sword if the character is within 2 units of her opponent. In which of the following scenarios would a "hit" be registered?

i) Player A $(1,4)$ and Player B $(2,2)$

ii) Player C $(-1,2)$ and Player D $(0,1)$

iii) Player E $(3,2)$ and Player F $(4,1)$

iv) Player G $(-2,-3)$ and Player H $\left(-1,0\right)$