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Published byNathan Webster Modified over 9 years ago
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Squaring a Number To square a number means to :
“Multiply it by itself” Example : means 9 x 9 = 81 means 10 x 10 = 100
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Squaring a Number Calculate: a) 2² b) 4² c) 8² d) 1² Find:
2 X 2 = 4 4 X 4 = 16 8 X 8 = 16 1 X 1 = 1 Find: a) 3² + 6² b) 4² + 1² c) 8² + 6² d) 9² + 5² 3 X X 6 4 X X 1 8 X X 6 9 X X 5 = 45 = 17 = 100 = 106
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Square Root of a number You now know how to find : 92 = 9 x 9 = 81
We can ‘undo’ this by asking “which number, times itself, gives 81” From the top line, the answer is 9 This is expressed as : “the SQUARE ROOT of 81 is 9” or in symbols we write :
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Square Root of a number Find: a) √36 b)√25 c) √1 d) √144 = 12 = 6 = 5
A = 49cm² This square has an area of 49cm². What is the length of one of the sides? √49 = 7 A = 4cm² This square has an area of 4cm². What is the length of one of the sides? √4 = 2
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Right – Angle Triangles
In a right angled triangle the side directly across from the right angle is called The Hypotenuse
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Right – Angle Triangles
Measure the length of a ? 3 4 Measure the length of b ? Complete the triangle and measure the length of c (the hypotenuse) 5
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Right – Angle Triangles
Measure the length of a ? 6 Measure the length of b ? 8 Complete the triangle and measure the length of C (the hypotenuse) 10
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Right – Angle Triangles
Measure the length of a ? 5 Measure the length of b ? 12 Complete the triangle and measure the length of c the hypotenuse 13
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Right – Angle Triangles
Can anyone spot a relationship between a2, b2, c2. a b c a2 b2 c2 3 4 5 9 16 25 12 13 144 169 6 8 10 36 64 100
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Pythagoras’s Theorem c b a
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Summary of Pythagoras’s Theorem
Note: The equation is ONLY valid for right-angled triangles.
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Pythagoras’s Theorem Calculate the lengths of the hypotenuse in each case c cm c cm 12cm 15cm 9cm 8cm 12cm 16cm c cm
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Calculating the Hypotenuse
Example 1 Q2. Calculate the longest length of the right- angled triangle below. c 8 12
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Calculating the Hypotenuse
Example 2 Q1. An aeroplane is preparing to land at Glasgow Airport. It is over Lennoxtown at present which is 15km from the airport. It is at a height of 8km. How far away is the plane from the airport? Aeroplane c b = 8 Airport a = 15 Lennoxtown
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Calculating the Hypotenuse
Example 1: Calculate the length of the missing side of this triangle. The Hypotenuse (longest side) 6cm a² + b² = c² 8² + 6² = c² = c² c² = 100 c = √100 = 10 8cm
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Calculating the Hypotenuse
Example 2: Calculate the length of the missing side of this triangle. 5cm The Hypotenuse (longest side) a² + b² = c² 12² + 5² = c² = c² c² = 169 c = √169 = 13 12cm
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Solving Real-Life Problems
When coming across a problem involving finding a missing side in a right-angled triangle, you should consider using Pythagoras’ Theorem to calculate its length. Example : A steel rod is used to support a tree which is in danger of falling down. What is the length of the rod? 15m 8m rod
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Solving Real-Life Problems
Example 2 A garden is rectangular in shape. A fence is to be put along the diagonal as shown below. What is the length of the fence. 10m 15m
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Length of the smaller side
To find the formula for calculating the length of a smaller side we have to re-arrange Pythagoras’ using the balancing method we learned in our Brackets and equation topic.
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Length of the smaller side
The Hypotenuse or Longest side What side of the triangle is “c” Length of the smaller side Pythagoras’ Theorem is : a² + b² = c² What if we want to find out a value for a? What we do to one side we have to do to the other a² + b² = c² -b² -b² a² = c² - b² Always ‘take away’ the shorter side from the longest side
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Length of the smaller side
The Hypotenuse or Longest side What side of the triangle is “c” Length of the smaller side Pythagoras’ Theorem is : a² + b² = c² What if we want to find out a value for b? What we do to one side we have to do to the other a² + b² = c² -a² -a² b² = c² - a² Always ‘take away’ the shorter side from the longest side
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Length of the smaller side
Always take small side away from hypotenuse when finding a sorter side!!! Example : Find the length of side a ? Check answer ! Always smaller than hypotenuse 20cm 12cm a cm
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Length of the smaller side
Always take small side away from hypotenuse when finding a sorter side!!! Example : Find the length of side b ? 10cm b cm 8 cm Check answer ! Always smaller than hypotenuse
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Length of the smaller side
Pythagoras’ Theorem is : a² + b² = c² What if we want to find out a value for a? a² + b² = c² We need to re-arrange Pythagoras’ Theorem to form a² = c² - b² Always ‘take away’ the shorter side from the longest side
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Length of the smaller side
Example : Find the length of side a ? a² = c² - b² a² = 13² - 12² a² = a² = 25 a = √25 = 5 13cm 12cm a cm
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Length of the smaller side
Example : Find the length of side b ? b² = c² - a² b² = 15² - 12² b² = b² = 81 a = √81 = 9 15cm b cm 12 cm
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Pythagoras Theorem Finding hypotenuse c Finding shorter side b c b a
Finding shorter side a
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