measurement AND geometry • Pythagoras and trigonometry
remember
1. The hypotenuse is the longest side of the triangle and is opposite the right-angle.
2. On your diagram check whether you are finding the length of the hypotenuse or one of
the shorter sides.
3. The length of the hypotenuse can be found if we are given the length of the two shorter
sides by using the formula: c2 = a2 + b2.
4. The length of the shorter side can be found if we are given the length of the hypotenuse
and the other shorter side by using the formula a2 = c2 − b2 or b2 = c2 − a2.
5. When using Pythagoras’ theorem always check the units given for each measurement.
6. If necessary, convert all measurements to the same units before using the rule.
7. Worded problems can be solved by drawing a diagram and using Pythagoras’ theorem
to solve the problem.
8. Worded problems should be answered in sentence form.
Exercise
6B
Individual
Pathways
Finding the length of sides
1 WE 1 For the following triangles, calculate the length of the hypotenuse, x.
a
b
c
eBook plus
x
3
x
5
Activity 6-B-1
Investigating side
lengths
doc-4079
Activity 6-B-2
Calculating side
lengths
doc-4080
24
2 WE 1 For each of the following triangles, calculate the length of the hypotenuse, giving
answers correct to 2 decimal places.
a
b
4.7
c
19.3
804
Tricky side-length
calculations
doc-4081
Digital doc
SkillSHEET 6.4
doc-6207
6.3
27.1
562
3 WE2 Find the length, correct to 1 decimal place, of the unmarked side in each of the
following triangles.
a
b
Digital doc
SkillSHEET 6.5
doc-6208
c
14
3.2
10
8.4
17
8
eBook plus
7
12
4
Activity 6-B-3
eBook plus
x
4 WE2 Find the value of the pronumeral, correct to 2 decimal places.
a
b
c
s
1.98
8.4
30.1
47.2
2.56
u
17.52
t
Chapter 6 Pythagoras’ theorem and trigonometry
173
measurement AND geometry • Pythagoras and trigonometry
5 Find the value of the pronumeral in each of the following triangles, correct to 2 decimal
places.
a
b
0.28
896
v
x
742
0.67
c
d
1.3
6.2
468
x
x
114
e
f
2870
w
1920
x
17.5
12.2
Understanding
  6 MC What is the length of the hypotenuse in this triangle?
A 25 cm
B 50 cm
C 50 mm
D 500 mm
E 2500 mm
7 MC What is the length of the third side in this triangle?
A 48.75 cm
B 0.698 m
C 0.926 m
D 92.6 cm
E 69.8 mm
4 cm
3 cm
82 cm
43 cm
  8 WE3 A right-angled triangle has a base of 50 mm and a height of 12 cm. Calculate the length
of the hypotenuse in mm.
  9 What is the length of the diagonal of a rectangle whose sides are:
a 10 cm and 8 cm?
b 620 cm and 400 cm?
c 17 cm and 3 cm?
10 WE4 A horse race is 1200 m. The track is straight, and 35 m wide. How much further than
1200 m will a horse run if it starts on the outside and finishes on the inside as shown?
Finishing
post
Starting gate
35 m
1200 m
11 WE4 A ladder leans against a vertical wall. The foot of the ladder is 1.2 m from the wall, and
the top of the ladder reaches 4.5 m up the wall. How long is the ladder?
174
Maths Quest 9 for the Australian Curriculum
measurement AND geometry • Pythagoras and trigonometry
12 WE5 A ladder that is 7 metres long leans up
13
14
15
16
against a vertical wall. The top of the ladder reaches
6.5 m up the wall. How far from the wall is the foot
of the ladder?
A
kite is attached to a string 150 m long. Sam
holds the end of the string 1 m above the ground,
and the horizontal distance of the kite from Sam is
150 m
80 m as shown at right. How far above the ground is
the kite?
80 m
1m
F
ind the length of the hypotenuse of the following
right-angled triangles, giving the answer in the units
specified.
a Sides 456 mm and 320 mm, hypotenuse
in cm.
b Sides 12.4 mm and 2.7 cm, hypotenuse
in mm.
c Sides 32 m and 4750 cm, hypotenuse
in m.
d Sides 2590 mm and 1.7 m, hypotenuse
in mm.
T
wo sides of a right-angled triangle are given. Find the third side
in the units specified. The diagram shows how each triangle is to
c
a
be labelled.
Remember: c is always the hypotenuse.
a a = 37 cm, c = 180 cm, find b in cm.
b
b a = 856 mm, b = 1200 mm, find c in cm.
c b = 4950 m, c = 5.6 km, find a in km.
d a = 125 600 mm, c = 450 m, find b in m.
Find the value of the pronumeral, correct to 2 decimal places for each of the following.
a
b
25
c
3x
2x
3x
4x
18
x
30
6x
Reasoning
17 A rectangular park is 260 m by 480 m. Danny usually trains by running 5 circuits around the
edge of the park. After heavy rain, two adjacent sides are too muddy to run along, so he runs a
triangular path along the other two sides and
the diagonal. Danny does 5 circuits of this
path for training. Show that Danny runs about
970 metres less than his usual training session.
18 An adventure water park has hired Sally to
build part of a ramp for a new water slide. She
builds a ramp which is 12 m long and rises to
a height of 250 cm. To meet the regulations,
the ramp must have a gradient between 0.1
and 0.25. Show that the ramp Sally has built is
within the regulations.
Chapter 6 Pythagoras’ theorem and trigonometry
175