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Malcolm Findlay from Madras College in St Andrews, Scotland has solved the first part of this problem:
Izumi Tomioka, Carol Chow and Priscilla Luk from The Mount School in York solved the second part of the problem:
The original hexagon has sides of length 3 units and we need to work out x, the lengths of the sides of the smaller hexagons.
The hexagon below has been made from 6 of the shaded triangles
above.
We need to find the length of one of the sides of this hexagon.
Splitting the shaded triangle into half gives us a right angled triangle such that: \cos30 = {1.5\over H}
Andrei Lazanu (aged 12) from School 205 in Bucharest, Romania, solved both parts of the problem.
This is how he tackled the second part:
To calculate the lengths of the sides of the smaller hexagons I used the following notations:
l for the length of side of the great hexagon
a for the length of side of the small hexagon
I used the following notation:
Triangle ACE is equilateral, because its sides are
congruent.
So, angle EAC is 60 °.
Angle FAB is 120 °, since each angle of a regular hexagon is 120 °.
Triangle AEF is congruent with triangle ACB, having all sides
congruent. They are also isosceles triangles.
This means that each of the angles FAE and CAB is
30 °.
Therefore angle EAB is 90 °.
Triangle AMN is also equilateral, because it has a
60 ° angle (MAN) and AM = AN.
Triangle ANB is isosceles, so AN and NB are congruent.
Therefore, in the right angled triangle MAB,
MA = MN = NB = a
AB has length l
Applying the Pythagorean Theorem: l^2 = (a + a)^2 - a^2
If the length of the side of the great hexagon is 3 units long l = 3
Therefore a = \sqrt{3} units
An alternative way of calculating "a" takes into account the
first part of the problem:
the area of the great hexagon is three times the area of the small
one.
For a hexagon of side l, the area is calculated as 6 times the
area of an equilateral triangle of side l,
this means that the area of the great hexagon is:
6{l^2 \sqrt3\over 4} = {l^2 3\sqrt3\over2} | $ | (1)$ |
The 3 smaller hexagons of side "a" have a total area of:
3{a^2 3\sqrt3\over 2} = {a^2 9\sqrt3\over2} | (2) |
Since, (1) and (2) represent the same area, 3a^2 = l
which is the same as we found above.
If the sides of the triangle in the diagram are 3, 4 and 5, what is the area of the shaded square?
Draw all the possible distinct triangles on a 4 x 4 dotty grid. Convince me that you have all possible triangles.