E8 Students are expected to
make generalizations about the
diagonal properties of squares
and rectangles and apply these
properties
What do we mean by the
‘diagonal’ of a two-dimensional
figure?
• Use a Mathematics Dictionary to find
the definition for the term ‘diagonal’.
Record this definition.
Investigate the diagonals of squares by completing
the activities below for each of the squares given to
you.
•How do the lengths of the two diagonals of your squares
compare?
•Check by measuring them. Share your findings.
•How do the diagonals of the squares appear to intersect?
•Check your prediction by measuring. Share your findings.
•What do the diagonals appear to do to each vertex angle?
•What do you think the measure of each of the angles at a
vertex is?
•Check by measuring. Share your findings.
•Next, cut out the four triangles made by the two diagonals.
Describe and compare these triangles.
What did you learn about the diagonals of a square during
this investigation?
The Diagonals of a Square:
•are equal in length
•bisect each other
•intersect to form four right angles and combined
with the previous properties this means they
are perpendicular-bisectors of each other
•are bisectors of the vertex angles of the square,
thus forming 45 degree angles
•form four congruent isosceles right triangles
Investigate the diagonals of squares by completing the
activities below for each of the squares given to you.
•How do the lengths of the two diagonals of the rectangle
compare?
•Check by measuring them. Share your findings.
•How do the diagonals of the rectangle appear to intersect?
•Check your prediction by measuring. Share your findings.
•What do the diagonals appear to do to each vertex angle?
•What do you think the measure of each of the angles at a
vertex is?
•Check by measuring. Share your findings.
•Next, cut out the four triangles made by the two diagonals.
Describe and compare these triangles.
What did you learn about the diagonals of a rectangle
during this investigation?
The Diagonals of a Rectangle:
•are equal in length
•bisect each other
•form two pairs of equal opposite angles at the point
of intersection
•form two angles at each vertex of the rectangle
that sum to 90 degrees and have the same measures
as the two angles at the other vertices
•form two pairs of congruent isosceles triangles
• E8.1 Draw squares that have diagonals of
length 8cm. What properties of a square did
you use to do this? Did everyone draw the
same square?
• E8.2 Cut a square along both diagonals.
Investigate the different shapes you can
make (i) using two of the triangles formed if
equal sides must abut, (ii) using three of the
triangles, and (iii) using all four triangles.
• E8.3 Draw rectangles that have diagonals
that intersect to form 60 degree angles.
Did everyone in the class draw the same
one? How do all the rectangles compare?
• E8.4 When the diagonals are drawn in a
rectangle, how do you know that each
triangle formed is ¼ of the rectangle?
• E8.5
Draw an isosceles right triangle.
Use a mira to draw the square for which
the triangle is one-quarter.
• E8.6 Draw a segment 12cm long. Use only
a mira to construct the square that has
this as a diagonal.
• E8.7
All triangles are rigid while
rectangles are not. One or both diagonals
are often used in the real world to make a
rectangular shape rigid. Explain what this
means and give a real world example.
• E8.8
Draw isosceles triangles.
Explain
how you could use these triangles to
construct rectangles that would have the
isosceles triangles represent one-quarter of
their areas.
E8.9 Imagine a family of rectangles has a
perimeter of 38cm and all of their sides
are whole numbers. Draw this family of
rectangles on graph paper. Which family
member has the greatest area?
The
longest diagonal?
• E8.10 Draw rectangles and show the two
diagonals. Measure one angle at a vertex
and one angle at the centre. Then find the
measures of all the other angles in the
figure
using
only
these
two
angle
measurements and your knowledge of
properties.