Solving triangles

1
Solving triangles
2
Your two triangles are the same shape and size
They are congruent
3
These two triangle have the same shape but
are different sizes
They are similar triangles
They have the same angles
4
Problems involving triangles can be solved by
scale drawing
See examples
Problems involving triangles can be solved by
calculation using geometry
See examples
Problems involving triangles can be solved by
calculation using trigonometry
See examples
5
Scale drawings are similar to the real thing
6
North
The map shows the journey of a ship.
Island
The ship leaves the Port, sails 30 miles North,
then 20 miles East.
MAP
N
Make a scale drawing produce a similar diagram.
I
How far is the Island from the port ?
?
Port
DRAWING
P
7
Geometry uses rules that are true for all
triangles
8
What is the size of the green angle in this
triangle ?
50º
Answer
55º
75º
Use Geometry
9
Pythagoras of Samos
Born about 569 BC in Samos, Ionia Died about
475 BC
10
Pythagoras Theorem is about triangles
Right angled triangles
Draw a square onto each side
The area of the largest square
equals
the area of the two smaller squares
See Examples
11
A
Calculate the length of side AB
x cm
6 cm
B
C
8 cm
By Pythagoras theorem
x2 6 2 82
36 64
100
x 10
The length of side AB is 10 cm
12
T
Calculate the length of side RT
y m
R
15 m
17 m
By Pythagoras theorem
17 2 y 2 15 2
289 y 2 225
y 2 64
S
y 8
The length of side RT is 8 m
13
Trigonometry uses facts about similar triangles
14
Try this experiment
Cut the other piece in half like this.
Take a sheet of A4 paper this way up. Cut it in
half
?
?
Repeat as often as you can so you get a sequence
of rectangles
15
Place all the rectangles on top of each other
like this
Draw a diagonal line
Cut along the diagonal line to make lots of right
angled triangles
?
Which triangles are congruent?
Which are similar?
16
All these right angled triangles are similar.
These triangles are the same shape but different
sizes.
Any triangle similar to these has the same angles
17
On card make some right angled triangles that
are twice as long as they are high
See if all your triangles are similar.
Measure the smallest angle
18
We can use trigonometry to compare similar
triangles
We can use scientific calculator to give us
information about right angles triangles of all
different shapes sizes.
19
H
To calculate the angle x
O
x
You need to label the sides H hypotenuse
(longest side, opposite the right angle) O
opposite (opposite the angle x) A adjacent
(next to the angle x)
A
Then you have to choose between these ratios
OH
sine x
(Oranges Have Segments)
AH
cosine x
(Apples Have Cores)
tangent x
OA
(Oranges Are Tasty)
20
To calculate the angle x
O
H
10 cm
x
Label the sides (H O A)
20 cm
A
Choose sin, cos or tan? The two sides involved
are O A
OA
(Oranges Are Tasty)
tan x
1020
tan x
tan x 0.5
Using the calculator tan-1 function
x 26.6º to 1 decimal place
21
H
To calculate the value of y
10 cm
y cm
Label the sides (H O A)
30º
O
Choose sin, cos or tan? The two sides involved
are O ( y cm) H (10 cm)
A
OH
(Oranges Have Segments)
sin 30º
Y 10
sin 30º
y 10 x sin 30º
Using the calculator sine function
x 5 cm
22
12 mm
A
To calculate the value of h
52º
O
Label the sides (H O A)
Choose sin, cos or tan? The two sides involved
are A H
h mm
H
AH
cos 52º
(Apples Have Cores)
12 h
cos 52º
12___ cos 52º
h
Using the calculator cosine function
h 19.5 mm to 3 significant figures