Pythag
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Trigonometry Pythagoras
J Dwen
2007
Page 1 of 6
J Dwen 2007
Trigonometry Pythagoras
J Dwen
2007
Page 2 of 6
Pythagoras Once upon a time there was this Greek geek called Pythagoras He lived long ago before calculators were invented (yes, that long ago!) He noticed that the tiles on his corridor floor that turned a slight corner made an interesting shape a right angled triangle with each side extended into a square made up of a whole number of tiles when he eliminated in his mind the extra tiles, he found if he added the number of tiles from the two small side squares they added up to the number of tiles in the big side square He took this thought to his living room and started drawing right-angled triangles and folding squares off the sides When he cut the parchment shapes up he found that he could always completely (and exactly) cover up the big square with the other two so long as he cut one of them into pieces to fit the gaps. because he couldn't check out every triangle he called his triangle law a theorem Pythagoras' Theorem In any right-angled triangle The square on the hypotenuse (Always the long side opposite the right angle) is the sum of the squares on the other two sides To work it out using a calculator First check that it’s a right angled triangle with the length of two sides shown and that you need the length of the third side 1 square everything 2 If you want a long side add... If you want a short side subtract 3 Square root to find the answer Summarized Pythagoras’ Theorem
( )2
+
√
Your Teacher may give you out some grid sheets like those on the next page If so you will be told what number of triangle to cut out Once you have cut out your triangle cut 3 squares as long as the sides on your triangle from the rest of the sheet. Once finished you should be able to arrange your cut outs like the triangle and squares above. The triangles have been chosen so the long side is an exact number of squares long. Put the biggest square on the bottom and the next biggest on top of that Cut the smallest square up (along the lines) to cover up the big square exactly. When finished you will have shown that, for your triangle, the square on the long side is equal to the squares on the other two sides added together. This is Pythagoras’ Theorem. We will use this theorem and a calculator to work out the lengths of the sides of right angled triangles.
Trigonometry Pythagoras
J Dwen
2
1
2
1
4
3
4
3
2007
Page 3 of 6
Trigonometry Pythagoras
J Dwen
2007
Page 4 of 6
Pythagoras’ Theorem Long Side Example Problem
Summarized Pythagoras’ Theorem
Find the long side
2
( )
+
-
X
11 m
√
8m
1
Square everything
( )2 Square Everything
X
112 = 121
82 = 64
2
+
-
If you need the long side Add If you need a short side Subtract
3
Need the long side So ADD
112 = 121
82 = 64
√
Square Root to find the answer
X
112 + 82 = 185
Square Root to find The Answer 2
11 = 121
X
112 + 82 = 185
√185 = 13.6 82 = 64
Answer…X = 13.6m
Trigonometry Pythagoras
J Dwen
2007
Page 5 of 6
Pythagoras’ Theorem Short Side Example Problem
Summarized Pythagoras’ Theorem
Find the short side
2
( )
+
-
12 m
X
√
7m
1
Square everything
( )2 Square Everything
122 = 144
X
72 = 49
2
+
-
If you need the long side Add If you need a short side Subtract
3
√
Square Root to find the answer
Need a short side So SUBTRACT
122 - 72 = 144 – 49 = 95
2
122 = 144
X 72 = 49
Square Root to find The Answer
2
12 - 7 = 95
√95 = 9.7
X
122 = 144
72 = 49
Answer…X = 9.7 cm
Trigonometry Pythagoras
J Dwen
2007
Page 6 of 6
Exercise 1 Find the length of the side without a length on it. Give the answers to 1dp. Remember to use Pythagoras Theorem Summaries Below
Summarized Pythagoras’ Theorem
( )2
Pythagoras' Theorem 1 2 3
Always give the units
Square everything If you want a long side add... If you want a short side subtract Square root to find the answer
+
√
Draw the diagram before you answer it. 1
4 2
7
20
8 10
6
11
6
9
17
3
12 5
3
7
34
13
7 11
17 20 9
11
10
27 17
8
12
19 25
8
18
14
20
13
14
11
15
4
13
16
14
6
15 16
17
J Dwen
2007
Page 1 of 6
J Dwen 2007
Trigonometry Pythagoras
J Dwen
2007
Page 2 of 6
Pythagoras Once upon a time there was this Greek geek called Pythagoras He lived long ago before calculators were invented (yes, that long ago!) He noticed that the tiles on his corridor floor that turned a slight corner made an interesting shape a right angled triangle with each side extended into a square made up of a whole number of tiles when he eliminated in his mind the extra tiles, he found if he added the number of tiles from the two small side squares they added up to the number of tiles in the big side square He took this thought to his living room and started drawing right-angled triangles and folding squares off the sides When he cut the parchment shapes up he found that he could always completely (and exactly) cover up the big square with the other two so long as he cut one of them into pieces to fit the gaps. because he couldn't check out every triangle he called his triangle law a theorem Pythagoras' Theorem In any right-angled triangle The square on the hypotenuse (Always the long side opposite the right angle) is the sum of the squares on the other two sides To work it out using a calculator First check that it’s a right angled triangle with the length of two sides shown and that you need the length of the third side 1 square everything 2 If you want a long side add... If you want a short side subtract 3 Square root to find the answer Summarized Pythagoras’ Theorem
( )2
+
√
Your Teacher may give you out some grid sheets like those on the next page If so you will be told what number of triangle to cut out Once you have cut out your triangle cut 3 squares as long as the sides on your triangle from the rest of the sheet. Once finished you should be able to arrange your cut outs like the triangle and squares above. The triangles have been chosen so the long side is an exact number of squares long. Put the biggest square on the bottom and the next biggest on top of that Cut the smallest square up (along the lines) to cover up the big square exactly. When finished you will have shown that, for your triangle, the square on the long side is equal to the squares on the other two sides added together. This is Pythagoras’ Theorem. We will use this theorem and a calculator to work out the lengths of the sides of right angled triangles.
Trigonometry Pythagoras
J Dwen
2
1
2
1
4
3
4
3
2007
Page 3 of 6
Trigonometry Pythagoras
J Dwen
2007
Page 4 of 6
Pythagoras’ Theorem Long Side Example Problem
Summarized Pythagoras’ Theorem
Find the long side
2
( )
+
-
X
11 m
√
8m
1
Square everything
( )2 Square Everything
X
112 = 121
82 = 64
2
+
-
If you need the long side Add If you need a short side Subtract
3
Need the long side So ADD
112 = 121
82 = 64
√
Square Root to find the answer
X
112 + 82 = 185
Square Root to find The Answer 2
11 = 121
X
112 + 82 = 185
√185 = 13.6 82 = 64
Answer…X = 13.6m
Trigonometry Pythagoras
J Dwen
2007
Page 5 of 6
Pythagoras’ Theorem Short Side Example Problem
Summarized Pythagoras’ Theorem
Find the short side
2
( )
+
-
12 m
X
√
7m
1
Square everything
( )2 Square Everything
122 = 144
X
72 = 49
2
+
-
If you need the long side Add If you need a short side Subtract
3
√
Square Root to find the answer
Need a short side So SUBTRACT
122 - 72 = 144 – 49 = 95
2
122 = 144
X 72 = 49
Square Root to find The Answer
2
12 - 7 = 95
√95 = 9.7
X
122 = 144
72 = 49
Answer…X = 9.7 cm
Trigonometry Pythagoras
J Dwen
2007
Page 6 of 6
Exercise 1 Find the length of the side without a length on it. Give the answers to 1dp. Remember to use Pythagoras Theorem Summaries Below
Summarized Pythagoras’ Theorem
( )2
Pythagoras' Theorem 1 2 3
Always give the units
Square everything If you want a long side add... If you want a short side subtract Square root to find the answer
+
√
Draw the diagram before you answer it. 1
4 2
7
20
8 10
6
11
6
9
17
3
12 5
3
7
34
13
7 11
17 20 9
11
10
27 17
8
12
19 25
8
18
14
20
13
14
11
15
4
13
16
14
6
15 16
17
