Pythagoras Theorem in Triangles
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be able to understand and use Pythagoras’ Theorem
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All
will
work together to understand Pythagoras’ Theorem and how it
relates to a triangle
Most
will work in pairs
to use Pythagoras Theorem to find missing sides
on a triangle
Rally coach
Some
will be able to use these skills
to solve written real
-
life problems
using Pythagoras Theorem and assess each other’s answers using
Carousal feedback
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3000 years ago the Egyptians used Pythagoras Theorem to build
the Great Pyramids using knotted rope to make a 90
o
angle using a
3,4,5 triangle. Today builders using pieces of wood with length 3ft,
4ft, 5ft to the same thing to get a perfect 90
o
right angle
Now you are going to try to find out
Now you are going to try to find out
what I discovered!!
what I discovered!!
I was a Mathematician who became
I was a Mathematician who became
famous for discovering something unique
famous for discovering something unique
about
about
right – angled triangles
right – angled triangles
.
.
12cm
5cm
8cm
6cm
1
2
3
4
c
12cm
5cm
8cm
6cm
1
2
3
4
c
a
a
2
2
+ b
+ b
2
2
= c
= c
2
2
3cm
4cm
x
7cm
9cm
x
x
= 11.4cm
= 11.4cm
x
2
2
= 130
= 130
9
9
2
2
= 81
= 81
7
7
2
2
= 49
= 49
4cm
8cm
x
x
= 8.9
= 8.9
x
2
2
= 80
= 80
8
8
2
2
= 64
= 64
4
4
2
2
= 16
= 16
12cm
7cm
x
x
= 9.7cm
= 9.7cm
x
2
2
= 95
= 95
12
12
2
2
= 144
= 144
7
7
2
2
= 49
= 49
23mm
15mm
x
x
= 17.4cm
= 17.4cm
x
2
2
= 304
= 304
23
23
2
2
= 529
= 529
15
15
2
2
= 225
= 225
For each of the following triangles, calculate the length of the missing side,
For each of the following triangles, calculate the length of the missing side,
giving your answers to one decimal place when needed.
giving your answers to one decimal place when needed.
Answer = 6.7cm
Answer = 6.7cm
Answer = 9.8cm
Answer = 9.8cm
Answer = 23.6m
Answer = 23.6m
Answer = 5mm
Answer = 5mm
Answer = 1.0cm
Answer = 1.0cm
Answer = 65cm
Answer = 65cm
Answer = 8.5cm
Answer = 8.5cm
Answer = 16.1cm
Answer = 16.1cm
Calculate the length of the
Calculate the length of the
diagonal of this square.
diagonal of this square.
6cm
If a right angle has short lengths
If a right angle has short lengths
14cm and 8cm, what is the
14cm and 8cm, what is the
length of the longest side.
length of the longest side.
12cm
12cm
8cm
Calculate the base of this
Calculate the base of this
isosceles triangle.
isosceles triangle.
10cm
10cm
8cm
Calculate the height of this
Calculate the height of this
isosceles triangle.
isosceles triangle.
Answer = 11.3cm
Answer = 11.3cm
Answer = 12cm
Answer = 12cm
•
Each team-mate has a different real – life problem.
Each team-mate has a different real – life problem.
•
On your own solve each of these problems.
On your own solve each of these problems.
•
Once you completed, swap with the other pupils on your table and give
Once you completed, swap with the other pupils on your table and give
feedback each others answers
feedback each others answers
A boat travels 45 miles east then 60 miles north, how far is it from where it
started? (hint: draw a diagram)
A swimming pool is 25m by 12m, if someone swam from one corner to the other,
how far would they have swam?
(hint: draw a diagram)
Answer = 75miles
Answer = 75miles
Answer = 27.7m
Answer = 27.7m
A ladder which is 4m long leans against a wall, the bottom of the ladder is
1.5m from the bottom of the wall, how high up the wall does the ladder go?
(hint: draw a diagram)
A rope of length 10m is stretched from the top of a pole 3m high until it
reaches the ground. How far is the end of the rope to the base of the
pole.(hint: draw a diagram)
Answer = 3.7m
Answer = 3.7m
Answer = 9.5m
Answer = 9.5m
Delve into the fascinating world of Pythagoras Theorem and its application in right-angled triangles. Uncover the historical significance of this mathematical concept, engage in practical activities to understand its principles, and discover the connection between the squares of the triangle's sides. Through hands-on exploration and real-life problem-solving, grasp the essence of Pythagoras Theorem and how it unlocks the mysteries of geometric relationships.
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Presentation Transcript
Learning Outcomes Using Pythagoras Theorem CONTENT- By the end of the lesson we will be able to understand and use Pythagoras Theorem PROCESS- We will know we are successful All will work together to understand Pythagoras Theorem and how it relates to a triangle Most will work in pairs to use Pythagoras Theorem to find missing sides on a triangle Rally coach Level 7 Some will be able to use these skills to solve written real-life problems using Pythagoras Theorem and assess each other s answers using Carousal feedback BENEFITS- We are learning this because 3000 years ago the Egyptians used Pythagoras Theorem to build the Great Pyramids using knotted rope to make a 90o angle using a 3,4,5 triangle. Today builders using pieces of wood with length 3ft, 4ft, 5ft to the same thing to get a perfect 90oright angle Level 8
Pythagoras Theorem I was born at Samos, in Greece, and lived from 580 to 500 B.C. I was a Mathematician who became famous for discovering something unique about right angled triangles. Now you are going to try to find out what I discovered!!
Pythagoras Theorem Make accurate copies of the three right-angled triangles below 3 1 2 3cm 6cm 5cm 4cm 12cm 8cm Next measure the length of the longest side of each one. Then complete the table below a2 b2 c2 a b c c 3 4 a 4 5 12 b 6 8 Can you see a pattern in the last 3 columns? . If you can then you have rediscovered Pythagoras Theorem
Pythagoras Theorem Make accurate copies of the three right-angled triangles below 3 1 2 3cm 6cm 5cm 4cm 12cm 8cm Next measure the length of the longest side of each one. Then complete the table below a2 9 25 36 b2 16 144 64 c2 25 169 100 a b c 5 13 10 c 3 4 a 4 5 12 b 6 8 Can you see a pattern in the last 3 columns? a2 + b2 = c2
Using Pythagoras Theorem So what is Pythagoras Theorem? Area C He said that: c2 Area A c a2 + b2 = c2 a a2 b Area B For any right triangle, the sum of the areas of the two small squares is equal to the area of the larger. b2 Pythagoras
Using Pythagoras Theorem We can use Pythagoras Theorem to find the longest side in a right angled triangle Area C Find the Length of side x 9 +16 = 25 Area A We SQUARE to get the area of the smaller squares x 3cm 32 = 9 4cm x = 25 = 5cm We ADD to get the area of the biggest square Area B How do we get the length of side x 42 = 16 We SQUARE ROOT the area to get the length of side x
Using Pythagoras Theorem Level 7 We can use Pythagoras Theorem to find the longest side in a right angled triangle Example 1 Find the Length of side x 92 = 81 72 = 49 x 1. Square 7cm 9cm x2 = 130 2. Add 3. Square x= 130 Root x = 11.4cm
Using Pythagoras Theorem Level 7 We can use Pythagoras Theorem to find the longest side in a right angled triangle Example 2 Find the Length of side x 82 = 64 42 = 16 x 1. Square 4cm 8cm x2 = 80 2. Add 3. Square x= 80 Root x = 8.9
Using Pythagoras Theorem Level 7 We can use Pythagoras Theorem to find a Short side in a right angled triangle Example 3 Find the Length of side x 12cm 122 = 144 72 = 49 1. Square x 7cm x2 = 95 2. Subtract 3. Square x= 95 Root x = 9.7cm
Using Pythagoras Theorem Level 7 We can use Pythagoras Theorem to find a Short side in a right angled triangle Example 4 Find the Length of side x 23mm 232 = 529 152 = 225 1. Square 15mm x x2 = 304 2. Subtract 3. Square x= 304 Root x = 17.4cm
Using Pythagoras Theorem For each of the following triangles, calculate the length of the missing side, giving your answers to one decimal place when needed. 19m 3 2 1 6cm 14m 5cm 3cm Answer = 23.6m Answer = 9.8cm Answer = 6.7cm 6 5 4 60cm 1.1cm 12mm 25cm Answer = 65cm Answer = 1.0cm Answer = 5mm Level 7 I have met todays Learning Outcome :
Using Pythagoras Theorem 8 7 If a right angle has short lengths 14cm and 8cm, what is the length of the longest side. Calculate the length of the diagonal of this square. 6cm Answer = 16.1cm Answer = 8.5cm 10 Calculate the base of this isosceles triangle. 9 Calculate the height of this isosceles triangle. Answer = 12cm 10cm 10cm Answer = 11.3cm 12cm 12cm 8cm 8cm I have met todays Learning Outcome :
Pythagoras Theorem Each team-mate has a different real life problem. On your own solve each of these problems. Once you completed, swap with the other pupils on your table and give feedback each others answers Real Life Problem 1 A boat travels 45 miles east then 60 miles north, how far is it from where it started? (hint: draw a diagram) Level 8 Answer = 75miles Real Life Problem 2 A swimming pool is 25m by 12m, if someone swam from one corner to the other, how far would they have swam?(hint: draw a diagram) Answer = 27.7m
Pythagoras Theorem Level 8 Real Life Problem 3 A ladder which is 4m long leans against a wall, the bottom of the ladder is 1.5m from the bottom of the wall, how high up the wall does the ladder go? (hint: draw a diagram) Answer = 3.7m Real Life Problem 4 A rope of length 10m is stretched from the top of a pole 3m high until it reaches the ground. How far is the end of the rope to the base of the pole.(hint: draw a diagram) Answer = 9.5m
