Upload
eric-george
View
222
Download
1
Tags:
Embed Size (px)
Citation preview
PYTHAGORAS
Aim: To be able to know Pythagoras’ Theorem
All: Will be able to recall theorem.Most: Will be able to use to find the length of hypotenuse.Some: Will be able to use it to find the length of the shorter side.
Study Plus
Pythagoras’ TheoremTrigonometry
Polygon anglesRatio & Proportion
Pythagoras’ Theorem
Only works in right angled triangles Nothing to do with angles Two main types
of questions Type 1 Type 2
Hypotenuse
Hypotenuse
The hypotenuse is the longest side in a right angled triangle.
It is always the side opposite the right angle.
hypotenuse
Spotting the Hypotenuse
hypotenuse
Pythagoras’ Theorem ‘In a right-angled triangle, the area of the square on
the side opposite to the right angle is equal to the sum of the squares on the sides forming the right angle.’
Pythagoras’ Theorem
Pythagoras’ Theorem states that: ‘The sum of the squares of the lengths of the sides
containing the right angle is equal to the square of the hypotenuse.’
In other words:
a2 + b2 = c2
a
b
c
A
BC
a2 + b2 = c2 ‘c’ must be the hypotenuse
You must square the numbers first, and then add
Remember that ‘square’ means to multiply the number by itself (32 = 3x3 = 9)
Type 1 (Finding The Hypotenuse)
a² + b ² Square, square Add Square root 10
18
?
Find the missing side.
102 = 100, 182 = 324
100 + 324 = 424
424 = 20.6 (3 sf)
Type 1Find the missing sides.
Give your answers to 3 sf.8c
m
10cm
11m
7m
24km
5km
102 = 100, 82 = 64100 + 64 = 164164 = 12.8cm
112 = 121, 72 = 49121 + 49 = 170170 = 13.0m
242 = 576, 52 = 25576 + 25 = 601601 = 24.5km
Type 2 (Finding A Leg)
Square, square Take away Square root
Find the missing side.
3.12 = 9.61, 22 = 4
9.61 – 4 = 5.61
5.61 = 2.37 miles (3 sf)
2 miles
3.1 miles
?
Type 2Find the missing sides.
Give your answers to 3 sf.9c
m
15cm
13m6m
24km
9km
152 = 225, 92 = 81225 – 81 = 144
144 = 12.0cm
132 = 169, 62 = 36169 – 36 = 133
133 = 11.5m
242 = 576, 92 = 81576 – 81 = 495
495 = 22.2km
Navigation (1)
Navigation problems are often solved using Pythagoras’ Theorem.
Make sure you know which way North, South, East and West point!
N
S
EW
Navigation (2)A plane leave an airport and travels 32km west then it turns and travels 41km north. It develops a problem and has to return to the airport. How far is it?
Step 1. Draw a diagram
32km Airport
?
Step 2. Use Pythagoras
This is Type 1. We have to find the hypotenuse.
41km 322 = 1024, 412 =
16811024 + 1681 = 27052705 =
52.0km
Word Problems (1)
Sometimes it is not obvious that you need to use Pythagoras’ Theorem.
If you draw a diagram you might spot a right angled triangle you can use…
Word Problems (2)Farmer Giles wants to cross to the diagonally opposite corner of his rectangular marrow field. The field measures 400m by 500m. How much distance does he save by going across the field rather than going around it?
Step 1. Draw a diagram
500m
400m?
Word Problems (3)Step 2. Use Pythagoras
400
500
5002 = 250 000, 4002 = 160 000
250 000 + 160 000 = 410 000
410000 = 640.3m
640.3
Step 3. The final answer
Distance round outside = 500 + 400 = 900m.
So Farmer Giles saves 900 – 640.3 = 260m (3 sf)
Word Problems (4)A ladder rests against a wall. For safety reasons the base of the ladder must be at least 2m from the wall. The ladder is 6.2m long. How high up the wall can the ladder reach?
Step 1. Draw a diagram
?
2m
6.2m
Word Problems (5)Step 2. Use Pythagoras
2m
5.87
m This is a Type 2 problem.
6.22 = 38.44, 22 = 4
34.44 = 5.87m (3 sf)
38.44 – 4 = 34.44
6.2m
And finally ….
Pythagoras’ Theorem only works for a particular type of triangle, which type?
If you are finding the hypotenuse, do you add or subtract the shorter sides squared?
What is meant by “hypotenuse”?
I wish I had worked
harder at school!