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Isosceles
Prove that a triangle with sides of length 5, 5 and 6 has the same area as a triangle with sides of length 5, 5 and 8. Find other pairs of non-congruent isosceles triangles which have equal areas.
Linkage
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?
Age 11 to 14
Challenge Level
Solutions received gave part of an insight into the problem but ommitted to consider triangles on the dotty grid that do not have a base of whole unit length. It is possible to draw triangles whose bases and heights are neither horizontal or vertical. These triangles have bases that are not a whole number or units. A complete solution needs to consider these.
Here is a synopsis of the solutions offered for the cases considered so far (i.e. it does not consider triangles that have non-horizontal bases):
The smallest triangle it is possibkle to draw has a base of 1 unit and a height of 1 unit. So the smallest area is $ \frac{1}{2} $ sq. unit.
There are an infinite number of triangles that can be drawn with these diagonals (see the problem "Shear Magic" )There are two ways of creating a triangle of area 1 sq and with a horizontal base:
Base 1 unit; height 2 units
or
Base 2 units and height 1 unit, again
For an area of 2 sq units there are three families of triangles with a hoirizontal base::
Base 1 unit and height 4 units
or
Base 2 units and height 2 units
or
Base 4 units and height 1 unit
For each family there are an infinite number of triangles
It is possible to demonstrate that it is also possible to obtain triangles with multiples of half a sq unit but there is still work to do.
NRICH is part of the family of activities in the Millennium Mathematics Project.