A circle touches the lines OA, OB and AB where OA and OB are
perpendicular. Show that the diameter of the circle is equal to the
perimeter of the triangle
A 1 metre cube has one face on the ground and one face against a
wall. A 4 metre ladder leans against the wall and just touches the
cube. How high is the top of the ladder above the ground?
The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?
The solution here depends on using
Pythagoras theorem in 3 dimensions (actually using the theorem for
a right angled triangle on the floor and then a second time for a
vertical right angled triangle). The centre of the large ball (of
radius $5$ cm) is at the centre of the box and, if you think of the
straight line from one corner of the box to the centre of the box,
it goes through the centre of a small `packing' ball.
This solution came from
Bishop's Stortford College.