### Degree Ceremony

What does Pythagoras' Theorem tell you about these angles: 90°, (45+x)° and (45-x)° in a triangle?

### Ball Bearings

If a is the radius of the axle, b the radius of each ball-bearing, and c the radius of the hub, why does the number of ball bearings n determine the ratio c/a? Find a formula for c/a in terms of n.

### After Thought

Which is larger cos(sin x) or sin(cos x) ? Does this depend on x ?

# Three by One

##### Age 16 to 18 Challenge Level:

$ABCD$ is a rectangle where $BC$ = $3AB$. $P$ and $Q$ are points on $BC$ such that $BP$ = $PQ$ = $QC$.

Show that: angle $DBC$ + angle $DPC$ = angle $DQC$

Generalise this result.

N.B. This problem can be tackled in at least 8 different ways using different mathematics learnt in the last two years in school and earlier. The methods are essentially the same when viewed from a more advanced perspective.