Enrichment in Mathematics – Shape, Space and Measures
Parallel Squares
Teachers Notes
This could be used as an additional or extension activity.
Without the ‘proof’ element it could be used as a starter or plenary.
It could be used to practice constructions, each pupil starting with their own
parallelogram, constructing the squares and midpoints, then ‘discovering’ the
property of the quadrilateral.
Pupils will benefit from working in pairs.
Geometrical software would help with trials of constructions.
(see Geogebra file for dynamic version)
Whether pupils use a vector or properties approach they will need to label each
element clearly and will benefit from having several copies of the diagram!
Solution
Vector proof:
A
D
NB = OM = b
NQ is the same length but perpendicular
so say NQ = b'
b
O
a
C
MB = ON = a
MP = a'
N
M
W
b
a'
P
B
b'
Q
OQ = ON + NQ = a + b'
OP = OM + MP = b + a'
So OQ is equal in length but perpendicular to OP.
Similarly the other pairs will be equal in length but perpendicular.
Therefore the diagonals are equal in length but perpendicular
so the shape is a square.
Worcestershire Numeracy Team
Enrichment Activities
Enrichment in Mathematics – Shape, Space and Measures
Geometric proof:
x
½x
½p
x
p
½x
½q
q
½x
Diagonal top to bottom = √[(½x+p+½x) ²+q²] = √[(x+p) ²+q²]
Diagonal left to right = √[(½p+x+½p) ²+(½q+ ½q²] = √[(x+p) ²+q²]
Diagonals equal in length, so must be a square or a rectangle.
½q
½x
½p
½x
½p
½q
Angle of blue diagonal to vertical:
½q
½x
+
½p
Angle of green diagonal to horizontal:
½x+½p
½q
So diagonals are perpendicular,
so shape must be a square.
Worcestershire Numeracy Team
Enrichment Activities
Enrichment in Mathematics – Shape, Space and Measures
Parallel Squares
Draw a parallelogram.
Draw squares on each edge.
Join the centres of the four squares.
What do you know about this quadrilateral?
Worcestershire Numeracy Team
Enrichment Activities