Determine the total shaded area of the 'kissing triangles'.
ABCDEFGH is a 3 by 3 by 3 cube. Point P is 1/3 along AB (that is AP
: PB = 1 : 2), point Q is 1/3 along GH and point R is 1/3 along ED.
What is the area of the triangle PQR?
Four rods, two of length a and two of length b, are linked to form
a kite. The linkage is moveable so that the angles change. What is
the maximum area of the kite?
Several pupils from The Mount School in York attempted this
problem. Two pupils began to try to explain how they knew they had
found all the solutions. They said:
"If you've got a base of 1 unit and a height of 1 unit then
there are 3 triangles possible,
if you've a base of 1 and a height of 2 then there are another 3
possible triangles and
a base of 1 with a height of 3 gives another 3.
So you've got 9 triangles with a base of 1"
Here are two diagrams to illustrate this:
This is a good and convincing start - they made 27 triangles but
do not appear to have considered triangles whose bases are not
Can anyone develop these excellent beginnings? Perhaps the
students at The Mount School could put their ideas together to come
up with a more "complete" solution.
Solution to Triangular Tantaliser This solution is offered by
Year 8 pupils at Hethersett High School (Andrew, Chris H and Chris
Thank you for your efforts and I am pleased that you were being
systemmatic; thinking of each of a range of different types of
It is a good start but I still need some convincing! Some editors
are hard to please!! For instance, I am not sure you have all the
right angled triangles. How can you pursuade me? This certainly
gets us on our way!
In total we found 26 different triangles without rotating,
reflecting or translating. We did this in groups of different
We started off finding all the right-angled traingles: there were
Then we found all the isosceles triangles, then all the irregular
triangles, giving us a total of 26.