GCSE Right-Angled Triangles
Dr J Frost (jfrost@tiffin.kingston.sch.uk)
Learning Objectives: To be able to find missing sides and missing
angles in right-angled triangles and 3D shapes.
Last modified: 2nd March 2014
RECAP: Pythagoras’ Theorem
!
Hypotenuse
(the longest side)
c
a
b
For any right-angled triangle
with longest side c.
a2 + b 2 = c 2
Example
Step 1: Determine the
hypotenuse.
x
2
Step 2: Form an equation
22
4
+
42
=
x2
The hypotenuse
appears on its own.
Step 3: Solve the equation to
find the unknown side.
x2 = 4 + 16 = 20
x = √20 = 4.47 to 2dp
Pythagoras Mental Arithmetic
We’ve so far written out the equation 𝑎2 + 𝑏 2 = 𝑐 2 , filled in our information,
and rearranged to find the missing side. But it’s helpful to be able to do it in our
heads sometimes!
If you’re looking for the hypotenuse  Square root the sum of the squares
If you’re looking for another side  Square root the difference of the squares
h
7
3
5
ℎ=
?
32 + 52 = 34
x
4
𝑥=
?
72 − 42 = 33
Pythagoras Mental Arithmetic
h
10
12
5
ℎ=
y
122 + 5? 2 = 13
9
𝑦=
x
92 − 22? = 77
102 − 4?2 = 84
q
1
2
2
𝑥=
4
𝑞=
22 + 1?2 = 5
The Wall of Triangle Destiny
Answer: 𝐱 = 𝟐?
2
Answer: 𝐱 = 𝟏𝟎?
3
42
1
1
1
5
6
x
x
x
6
4
x
55
x
12
8
Answer: 𝐱 = 𝟒𝟕𝟖𝟗
?
4
Answer: 𝐱 = 𝟐𝟎
?
10
Answer: 𝐱 = 𝟒𝟒
?
“To learn secret way of ninja,
find x you must.”
Exercise 1
Give your answers in both surd form and to 3 significant figures.
4
1
x
7
6
y
8
?
?
5
2
x
13
10
x = 65 = 13.4
x = 10
Find the height of
this triangle.
?
12
6
10
y
4
3
?
2
?
x = 29 = 5.39
9
7
1
6
5
x
?
x = 51 = 7.14
x
N
x = 43 = 6.56
7
3
13
18
12
x
1
x
x2 + 49 = 81 – x2
x=4
?
1
?
x = 3 = 1.73
Areas of isosceles triangles
To find the area of an isosceles triangle, simplify
split it into two right-angled triangles.
13
13
12
?
1
1
3
?
2
10
1
Area = 60
?
Area = 3
?
4
Exercise 2
Determine the area of the following triangles.
1
3
5
5
12
5
17
17
7
12
6
Area = 12
?
2
16
Area = 120
?
4
4
4
1
1
4
1.6
Area = 212
= 43?= 6.93
Area = 0.48
?
Area = 40.2
?
y
(a,b)
When I was in Year 9 I
was trying to write a
program that would
draw an analogue
clock.
I needed to work out
between what two
points to draw the
hour hand given the
current hour, and the
length of the hand.
θ
r
x
Trigonometry
Given a right-angled triangle, you know how to find a missing side
if the two others are given. But what if only one side and an angle
are given?
x
4
30°
y
Names of sides relative to an angle
hypotenuse
?
opposite
?
30°
adjacent
?
Names of sides relative to an angle
Hypotenuse
x
Opposite
Adjacent
60°
z
y
1
√2
45°
?x
?y
?z
?
√2
?1
?1
?c
?a
?b
1
c
20°
a
b
Sin/Cos/Tan
sin, cos and tan give us the ratio between pairs of sides in
a right angle triangle, given the angle.
𝑜
sin 𝜃 = ?
ℎ
h
o
θ
a
𝑎
cos 𝜃 = ?
ℎ
𝑜
tan 𝜃 = ?
𝑎
“soh cah toa”
Example
Looking at this triangle, how many
times bigger is the ‘opposite’ than
the ‘adjacent’ (i.e. the ratio)
Ratio is 1 (they’re?the same length!)
Therefore:
opposite
?
tan(45)
=1
45
adjacent
?
More Examples
Step 1: Determine which sides are hyp/adj/opp.
Step 2: Work out which trigonometric function we need.
Find 𝑥 (to 3sf)
20 °
7
4
40 °
x
𝑥 = 3.06
?
x
𝑥 = 2.39
?
More Examples
𝑥 = 24?
60 °
x
12
x
4
30°
𝑥=8 ?
Exercise 3
Find 𝑥, giving your answers to 3𝑠𝑓. Please copy the diagrams first.
1
b
a
𝑥 = 16.9
?
22
15
𝟕𝟎°
𝒙
𝟖𝟎°
𝟒𝟎°
𝑥 = 14.1
?
𝑥 = 20.3
?
c
𝒙
𝒙
e
d
𝟕𝟎°
4
𝑥 = 7.00
?
20
f
𝑥 = 11.0
?
𝟒
𝟕𝟎°
𝑥 = 11.7
?
𝒙
2
𝒙
I put a ladder 1.5m away from a tree. The ladder is inclined at 70° above the
horizontal. What is the height of the tree? 𝟒. 𝟏𝟐𝒎
?
3
Ship B is 100m east of Ship A, and the bearing of Ship B from Ship A is 30°. How far
North is the ship? 𝟏𝟎𝟎 ÷ 𝐭𝐚𝐧 𝟑𝟎 = 𝟏𝟕𝟑. 𝟐𝒎
?
4
Find 𝑥.
𝟑𝟎°
𝒙+𝟏
𝒙
𝒙=
𝟏
?
𝟑−𝟏
y
𝑎, 𝑏
θ
𝑟
x
So what is 𝑎, 𝑏 ?
RECAP: Find x
4
𝑥=
= 4? 3 𝑜𝑟 6.93
tan 30
4
30 °
x
But what if the angle is unknown?
5
3
𝒂
3
sin
? 𝑎 =
5
𝑆𝑜 𝑎 =
sin−1
3
= 36.9°
?
5
We can do the ‘reverse’ of sin, cos or tan to find the missing angle.
What is the missing angle?
𝟓
𝒂
𝟒
cos −1
4
5
cos−1
5
4
cos−1
4
5
sin−1
5
4
What is the missing angle?
𝟏
𝒂
𝟐
cos −1
1
2
sin−1
2
tan−1
2
tan−1
1
2
What is the missing angle?
𝟓
𝟑
𝒂
cos −1
3
5
sin−1
3
5
tan−1
3
5
sin−1
5
3
What is the missing angle?
𝟑
𝒂
𝟐
cos −1
2
3
sin−1
2
3
sin−1
3
2
tan−1
2
3
2
1
1
θ
2
1
1
θ
𝜃 = 48.59°
?
4
3
6
θ
3
𝜃 =?45°
3
𝜃 = 70.53°
?
8
θ
𝜃 =?
33.7°
“To learn secret way of
math ninja, find θ you must.”
3.19m
40°
x
Find x
60°
3m
Exercises
GCSE questions on provided worksheet
3D Pythagoras
The strategy here is to use Pythagoras twice, and use some internal
triangle in the 3D shape.
Determine the length of
the internal diagonal of
a unit cube.
1
?
√3
?
√2
1
Click to BroSketch
1
Test Your Understanding
The strategy here is to use Pythagoras twice, and use some internal
triangle in the 3D shape.
Determine the length of
the internal diagonal of
a unit cube.
12
?
13
4
3
Test Your Understanding
Determine the height of
this right* pyramid.
2
?
2
2
2
* A ‘right pyramid’ is one where the
top point is directly above the centre
of the base, i.e. It’s not slanted.
Exercise 4
Determine the length x in each diagram. Give your answer in both surd for and as a
decimal to 3 significant figures.
1
x
1
3
13
N1
2
x
2
3
x = 14 =?3.74
2
6
8
4
x = 45 =?6.71
x
2
2
2
x = 28 =?5.29
N2
x
8
5
2
2
2
x = 12?
4
x
2
4
x = 51 =?7.14
1
x
1
6
1
Hint: the centre of a triangle is 2/3 of the way along the
diagonal connecting a corner to the opposite edge.
x = (2/3) ?
= 0.816