What is the area of the quadrilateral APOQ? Working on the building
blocks will give you some insights that may help you to work it
Six circular discs are packed in different-shaped boxes so that the
discs touch their neighbours and the sides of the box. Can you put
the boxes in order according to the areas of their bases?
Given a square ABCD of sides 10 cm, and using the corners as
centres, construct four quadrants with radius 10 cm each inside the
square. The four arcs intersect at P, Q, R and S. Find the area
enclosed by PQRS.
The answer 1/6 was found in a rich
variety of ways, but what is really important is
not the answer but proving it.
Alice, Clare, Jenny, Anna, Tam, Claire and their mates in year 10,
from the Mount School, York (I hope
no-one has been left out) found that there were some
triangles that are similar. They had
begun "by trying to draw it accurately, using different
sizes of square, and those of us who began with a square of side 12
cm arrived at the answer first, because the points all land on easy
grid points and the numbers are easy. We then worked back trying to
prove/convince ourselves of what we were assuming!".
This group knew that, to make this a really
good solution they would have to explain why the vertices of the
octagon lie on 'easy' grid points, and they knew the reasons were
to do with similar triangles.
Here is the diagram drawn by
Amy of Madras
College . Her diagram shows clearly
that the octagon has an area of 24 square units and the large
square an area of 144 square units. Amy gave a good account of how
she worked out the answer, but again she assumed, but did not
prove, that the vertices of the octagon were exactly on those easy
grid points. No-one can know that just from drawing!
From the gradient of the line O
R , or the fact that triangles O
P L and O
R N are similar, it follows that
P L = 1/2 R
N which proves that the coordinates of
P are (6,3).
By symmetry Q is on the diagonal of the square
and Q M = M
N . Again either from the gradient of the line
O R , or the fact that triangles
O Q M and
O R N are
similar we know that O M =2
Q M and putting these two results
together we have O M =2
M N so M is the
point (8,0) and Q is the point (8,4).
communique, but without proof, was from Laura, Charlotte and Laura
of Maidstone Girls' Grammar School .