Alice and Bob want to get from one
corner of this rectangular field to the
other. Alice walks round the edge of the
field. Bob cuts right across. How much
further did Alice walk?
Pythagoras’ Theorem – Exercise 1
1. Find the side marked with the letter (you
do not need to copy the diagrams).
3. [Kangaroo Pink 2008 Q3] Four unit
squares are placed edge to edge as
shown. What is the length of the line 𝑃𝑄?

2. To rescue a cat I put a ladder of length
10m against a tree, with the foot of the
latter 2.5m away from the tree. How high
up the tree is the cat?
Pythagoras’ Theorem – Exercise 1
1. Find the side marked with the letter (you
do not need to copy the diagrams).
[Based on JMO 1996 A6] The
length of the shortest diagonal
of an octagon is 1. What is the
length of the longest diagonal?
3. Alice and Bob want to get from one
corner of this rectangular field to the
other. Alice walks round the edge of the
field. Bob cuts right across. How much
further did Alice walk?
4. [Kangaroo Pink 2008 Q3] Four unit
squares are placed edge to edge as
shown. What is the length of the line 𝑃𝑄?
2. To rescue a cat I put a ladder of length
10m against a tree, with the foot of the
latter 2.5m away from the tree. How high
up the tree is the cat?
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5. [Based on JMO 1996 A6] The
length of the shortest diagonal
of an octagon is 1. What is the
length of the longest diagonal?
Pythagoras’ Theorem – Exercise 2
5. (a) What is the height of an
equilateral triangle with side length 6?
Give your answer in exact form unless
otherwise specified.
1. Determine 𝑥.
(b) What is its area?
6. Determine 𝑧.
2. Determine 𝑦.
7. Find the height of this
isosceles triangle.
8. Find the area of this isosceles
triangle.
3. Santa Claus and Rudolph are sitting at
the corner of a square swimming pool
of 10m by 10m, which has frozen
over. The want to get to the other
corner of the pool, where Mary
Christmas has left some brandy and a
carrot. Rudolph runs around the edge
of the pool, while Santa, who has
recently been on a diet, decides to
risk walking diagonally across the ice.
Calculate the distance saving, to 2
decimal places.
4. Two snowmen are back to back,
facing in opposite directions. They
each walk 3km forward, turn left and
then work a further 4km. How far are
the snowmen from each other?
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9. Determine 𝑥.
10. [IMC 2008 Q20] What, in cm2, is the
area of this quadrilateral?
11. [IMC 2009 Q20] A square, of side two
units, is folded in half to form a
triangle. A second fold is made,
parallel to the first, so that the apex of
this triangle folds onto a point on its
base, thereby forming an isosceles
trapezium. What is the perimeter of
this trapezium?
A 4 + √2
B 4 + 2√2
C 3 + 2√2
D 2 + 3√2 E 5
12. Determine 𝑥 (using an algebraic
method).
 Killer 3: [JMO 2007 B5] A window is
constructed of six identical panes of
glass. Each pane is a pentagon with
two adjacent sides of length two
units. The other three sides of each
pentagon, which are on the perimeter
of the window, form half of the
boundary of a regular hexagon.
Calculate the exact area of the glass in
the window.
 Killer 4: [JMO Mentoring May2012
Q4] A triangle has two angles which
measure 30° and 105°. The side
between these angles has length 2
cm. What is the perimeter of the
triangle? (Hint: split the triangle
somehow?)
 Killer 5: Determine 𝑥 (using an
algebraic method).
 Killer 1: [JMO 2006 B4] Start with an
equilateral triangle 𝐴𝐵𝐶 of side 2
units, and construct three outwardpointing squares ABPQ, BCTU, CARS
and the three sides AB, BC, CA. What
is the area of the hexagon PQRSTU?
 Killer 2: [JMO 1999 B4] The regular
hexagon 𝐴𝐵𝐶𝐷𝐸𝐹 has sides of length
2. The point 𝑃 is the midpoint of 𝐴𝐵.
𝑄 is the midpoint of 𝐵𝐶 and so on.
Find the area of the hexagon
𝑃𝑄𝑅𝑆𝑇𝑈.
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 Killer 6: Determine 𝑥.