### Pythagorean Triples

How many right-angled triangles are there with sides that are all integers less than 100 units?

### Tennis

A tennis ball is served from directly above the baseline (assume the ball travels in a straight line). What is the minimum height that the ball can be hit at to ensure it lands in the service area?

### Square Pegs

Which is a better fit, a square peg in a round hole or a round peg in a square hole?

# Where Is the Dot?

##### Stage: 3 Challenge Level:

Watch the film below.

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Imagine the dot starts at the point $(1,0)$ and turns anticlockwise.

Estimate the height of the dot above the horizontal axis after it has turned through $45^\circ$.

Estimate the angle that the dot needs to turn in order to be exactly $0.5$ units above the horizontal axis.

Show how you can use Pythagoras' Theorem to calculate the height of the dot above the horizontal axis after it has turned through $45^\circ$.

Again, without resorting to Trigonometry, calculate the height of the dot above the horizontal axis after it has turned through $30^\circ$ and $60^\circ$?

Are there any other angles for which you can calculate the height of the dot above the horizontal axis?