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© Boardworks Ltd of 60 S2 Pythagoras’ Theorem KS4 Mathematics

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© Boardworks Ltd of 60 A A A A A A S2.1 Introducing Pythagoras’ Theorem Contents S2 Pythagoras’ Theorem S2.3 Pythagorean triples S2.6 Applying Pythagoras’ Theorem in 3-D S2.2 Identifying right-angled triangles S2.4 Finding unknown lengths S2.5 Applying Pythagoras’ Theorem in 2-D

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© Boardworks Ltd of 60 Right-angled triangles A right-angled triangle contains a right angle. The longest side opposite the right angle is called the hypotenuse.

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© Boardworks Ltd of 60 Identify the hypotenuse

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© Boardworks Ltd of 60 The history of Pythagoras’ Theorem Pythagoras’ Theorem concerns the relationship between the sides of a right angled triangle. Although the Theorem is named after Pythagoras the result was known to many ancient civilizations including the Babylonians, Egyptians and Chinese, at least 1000 years before Pythagoras was born. The theorem is named after the Greek mathematician and philosopher, Pythagoras of Samos.

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© Boardworks Ltd of 60 Pythagoras’ Theorem Pythagoras’ Theorem states that the square formed on the hypotenuse of a right-angled triangle … … has the same area as the sum of the areas of the squares formed on the other two sides.

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© Boardworks Ltd of 60 Showing Pythagoras’ Theorem

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© Boardworks Ltd of 60 Pythagoras’ Theorem If we label the length of the sides of a right-angled triangle a, b and c as follows, then the area of the largest square is c × c or c 2. a c b a2a2 The areas of the smaller squares are a 2 and b 2. c2c2 b2b2 We can write Pythagoras’ Theorem as c 2 = a 2 + b 2

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© Boardworks Ltd of 60 A proof of Pythagoras’ Theorem

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© Boardworks Ltd of 60 A A A A A A S2.2 Identifying right-angled triangles Contents S2 Pythagoras’ Theorem S2.3 Pythagorean triples S2.6 Applying Pythagoras’ Theorem in 3-D S2.1 Introducing Pythagoras’ Theorem S2.4 Finding unknown lengths S2.5 Applying Pythagoras’ Theorem in 2-D

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© Boardworks Ltd of 60 Pythagoras’ Theorem Pythagoras’ Theorem states that for a right-angled triangle with a hypotenuse of length c and the shorter sides of lengths a and b c 2 = a 2 + b 2 a b c We can use Pythagoras’ Theorem to check whether a triangle is right-angled given the lengths of all the sides, to find the length of a missing side in a right-angled triangle given the lengths of the other two sides.

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© Boardworks Ltd of 60 Identifying right-angled triangles If we are given the lengths of the three sides of a triangle, we can use Pythagoras’ Theorem to find out whether or not the triangle contains a right angle. For example, A triangle has sides of length 4 cm, 7 cm and 9 cm. Is this a right-angled triangle? If the sum of the squares on the two shorter sides is equal to the square on the longest side, the triangle has a right angle = = 9 2 No, this is not a right-angled triangle.

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© Boardworks Ltd of 60 Identifying right-angled triangles

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© Boardworks Ltd of 60 Identifying right-angled triangles If the sum of the squares on the two shortest sides of a triangle is less than the sum of the square on the longest side, all the angles in the triangle must be acute. In other words, if a triangle with sides of length a, b and c, where c is the longest side has, c 2 < a 2 + b 2 then all the angles in the triangle must be acute.

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© Boardworks Ltd of 60 Identifying right-angled triangles If the sum of the squares on the two shortest sides of a triangle is greater than the sum of the square on the longest side, one of the angles in the triangle must be obtuse. In other words, if a triangle with sides of length a, b and c, where c is the longest side has, c 2 > a 2 + b 2 then one of the angles in the triangle must be obtuse.

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© Boardworks Ltd of 60 Identifying right-angled triangles A triangle has sides of length 5 cm, 8 cm and 10 cm. Is the angle opposite the longest side acute, obtuse or right-angled? 10 cm 8 cm 5 cm ? = = = > 89 so the angle opposite the longest side is an obtuse angle.

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© Boardworks Ltd of 60 Obtuse, acute or right-angled triangle

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© Boardworks Ltd of 60 A A A A A A S2.3 Pythagorean triples Contents S2 Pythagoras’ Theorem S2.6 Applying Pythagoras’ Theorem in 3-D S2.1 Introducing Pythagoras’ Theorem S2.2 Identifying right-angled triangles S2.4 Finding unknown lengths S2.5 Applying Pythagoras’ Theorem in 2-D

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© Boardworks Ltd of 60 Pythagorean triples A triangle has sides of length 3 cm, 4 cm and 5 cm. Does this triangle have a right angle? Using Pythagoras’ Theorem, if the sum of the squares on the two shorter sides is equal to the square on the longest side, the triangle has a right angle = = 25 = 5 2 Yes, the triangle has a right-angle. The numbers 3, 4 and 5 form a Pythagorean triple.

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© Boardworks Ltd of 60 Pythagorean triples Ancient Egyptians used the fact that a triangle with sides of length 3, 4 and 5 contained a right-angle to mark out field boundaries and for building.

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© Boardworks Ltd of 60 Pythagorean triples Three whole numbers a, b and c, where c is the largest, form a Pythagorean triple if, a 2 + b 2 = c 2 3, 4, 5 is the simplest Pythagorean triple. Write down every square number from 1 2 = 1 to 20 2 = 400. Use these numbers to find as many Pythagorean triples as you can. Write down any patterns that you notice.

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© Boardworks Ltd of 60 Pythagorean triples How many of these did you find? = = 5 2 3, 4, = = , 8, = = , 12, = = ,12, = = , 15, = = , 16, 20 The Pythagorean triples 3, 4, 5;5, 12, 13and 8, are called primitive Pythagorean triples.

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© Boardworks Ltd of 60 ? ? Similar right-angled triangles The following right-angled triangles are similar Find the lengths of the missing sides Check that Pythagoras’ Theorem holds for both triangles.

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© Boardworks Ltd of 60 Similar right-angled triangles = = 100 = = = 225 = 15 2 ? ? The following right-angled triangles are similar

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© Boardworks Ltd of 60 ? ? Similar right-angled triangles The following right-angled triangles are similar. Find the lengths of the missing sides. Check that Pythagoras’ Theorem holds for both triangles.

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© Boardworks Ltd of 60 Similar right-angled triangles The following right-angled triangles are similar = = 4624 = = = 2601 = 51 2

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© Boardworks Ltd of ? ? Similar right-angled triangles The following right-angled triangles are similar. Find the lengths of the missing sides. Check that Pythagoras’ Theorem holds for both triangles

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© Boardworks Ltd of 60 Similar right-angled triangles The following right-angled triangles are similar = = 6.25 = = = 9 = 3 2

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© Boardworks Ltd of 60 A A A A A A S2.4 Finding unknown lengths Contents S2 Pythagoras’ Theorem S2.6 Applying Pythagoras’ Theorem in 3-D S2.3 Pythagorean triples S2.1 Introducing Pythagoras’ Theorem S2.2 Identifying right-angled triangles S2.5 Applying Pythagoras’ Theorem in 2-D

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© Boardworks Ltd of 60 Pythagoras’ Theorem Pythagoras’ Theorem states that for a right-angled triangle with a hypotenuse of length c and the shorter sides of lengths a and b c 2 = a 2 + b 2 a b c We can use Pythagoras’ Theorem to check whether a triangle is right-angled given the lengths of all the sides, to find the length of a missing side in a right-angled triangle given the lengths of the other two sides.

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© Boardworks Ltd of 60 Finding the length of the hypotenuse Use Pythagoras’ Theorem to calculate the length of side a. a 5 cm 12 cm Using Pythagoras’ Theorem, a 2 = a 2 = a 2 = 169 a = 169 a = 13 cm

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© Boardworks Ltd of 60 Finding the length of the hypotenuse Use Pythagoras’ Theorem to calculate the length of side PR. P 0.7 m 2.4 m Using Pythagoras’ Theorem PR 2 = PQ 2 + QR 2 Substituting the values we have been given, PR 2 = PR 2 = PR 2 = 6.25 PR = 6.25 PR = 2.5 m R Q

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© Boardworks Ltd of 60 Finding the length of the hypotenuse Use Pythagoras’ Theorem to calculate the length of side x to 2 decimal places. 3 cm 6 cm Using Pythagoras’ Theorem x 2 = x 2 = x 2 = 45 x = 45 x = 6.71 cm x

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© Boardworks Ltd of 60 Finding the length of the shorter sides Use Pythagoras’ Theorem to calculate the length of side a. a 10 cm 26 cm Using Pythagoras’ Theorem, a = 26 2 a 2 = 26 2 – 10 2 a 2 = 676 – 100 a 2 = 576 a = 576 a = 24 cm

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© Boardworks Ltd of 60 Finding the length of the shorter sides Use Pythagoras’ Theorem to calculate the length of side AC to 2 decimal places. Using Pythagoras’ Theorem AB 2 + AC 2 = BC 2 Substituting the values we have been given, AC 2 = 8 2 AC 2 = 8 2 – 5 2 PR 2 = 39 PR = 39 PR = 6.24 cm A 5 cm 8 cm C B

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© Boardworks Ltd of 60 Finding the length of the shorter sides Use Pythagoras’ Theorem to calculate the length of side x to 2 decimal places. 7 cm 15 cm x Using Pythagoras’ Theorem, x = 15 2 x 2 = 15 2 – 7 2 x 2 = 225 – 49 x 2 = 176 x = 176 x = cm

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© Boardworks Ltd of 60 Find the correct equation

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© Boardworks Ltd of 60 Complete this table

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© Boardworks Ltd of 60 A A A A A A S2.5 Applying Pythagoras’ Theorem in 2-D Contents S2 Pythagoras’ Theorem S2.6 Applying Pythagoras’ Theorem in 3-D S2.3 Pythagorean triples S2.1 Introducing Pythagoras’ Theorem S2.2 Identifying right-angled triangles S2.4 Finding unknown lengths

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© Boardworks Ltd of 60 Finding the lengths of diagonals Pythagoras’ Theorem has many applications. For example, we can use it to find the length of the diagonal of a rectangle given the lengths of the sides. Use Pythagoras’ Theorem to calculate the length of the diagonal, d cm 13.6 cm d d 2 = d 2 = d 2 = 289 d = 289 d = 17 cm

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© Boardworks Ltd of 60 Finding the lengths of diagonals Use Pythagoras’ Theorem to calculate the length d of the diagonal in a square of side length 7 cm. 7 cm Using Pythagoras’ Theorem d 2 = d 2 = d 2 = 98 d = 98 d = 9.90 cm (to 2 d.p.) d

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© Boardworks Ltd of 60 Finding the lengths of diagonals Show that the length of the diagonal d in a square of side length a can be found by the formula d = √2 a a Using Pythagoras’ Theorem d 2 = a 2 + a 2 d 2 = 2 a 2 d Use this formula to find the length of the diagonal in a square with side length 12 cm to 2 decimal places. d = 2a 2 d = 2 a

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© Boardworks Ltd of 60 Finding the height of an isosceles triangle Use Pythagoras’ Theorem to calculate the height h of this isosceles triangle. 8 cm h = h 2 = – 4 2 h 2 = – 16 h 2 = h = h = 4.2 cm 5.8 cm h Using Pythagoras’ Theorem in half of the isosceles triangle, we have 5.8 cm 4 cm h

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© Boardworks Ltd of 60 Finding the height of an equilateral triangle Use Pythagoras’ Theorem to calculate the height h of an equilateral triangle with side length 4 cm. h = 4 2 h 2 = 4 2 – 2 2 h 2 = 16 – 4 h 2 = 12 h = 12 h = 3.46 cm (to 2 d.p.) 4 cm h Using Pythagoras’ Theorem in half of the equilateral triangle, we have h 4 cm 2 cm

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© Boardworks Ltd of 60 Finding the height of an equilateral triangle Using Pythagoras’ Theorem in half of the equilateral triangle, we have Show that the height h in an equilateral triangle of side length a can be found by the formula h = √3 a a h 2 h a a 2 h 2 + = a 2 4 a2a2 h 2 = a 2 – 4 a2a2 h 2 = 4 3a23a2 h = a 2 33 We can think of this as 1 a 2 – a 2 1 4

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© Boardworks Ltd of 60 Ladder problem

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© Boardworks Ltd of 60 Flight path problem

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© Boardworks Ltd of 60 Applying Pythagoras’ Theorem twice Sometimes we have to apply Pythagoras’ Theorem twice to find a required length. For example, Find the length of side a. To solve this problem we need to find the length of the other missing side which we can call b. 4 cm9 cm 18 cm a b b 2 = 18 2 – (4 + 9) 2 b 2 = 324 – 169 b 2 = 155 Now, a 2 = b a 2 = a 2 = 171 a = cm (to 2 d.p.)

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© Boardworks Ltd of 60 Find the given lengths A(–4, 5) B(–4, –2) C (6, –2) What is length of the line AB? The points A(–4, 5), B(–4, –2), and C (6,–2) form a right angled triangle. AB = 5 – –2 = = 7 units 7 units

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© Boardworks Ltd of 60 Find the given lengths The points A(–4, 5), B(–4, –2), and C (6,–2) form a right angled triangle. A(–4, 5) B(–4, –2) C (6, –2) What is length of the line BC? BC = 6 – –4 = = 10 units 7 units 10 units

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© Boardworks Ltd of 60 Find the given lengths A(–4, 5) B(–4, –2) C (6, –2) What is length of the line AC? Using Pythagoras, AC 2 = AB 2 + AC 2 The points A(–4, 5), B(–4, –2), and C (6,–2) form a right angled triangle. = = = 149 AC = 149 = units (to 2 d.p.) 7 units 10 units units (to 2 d.p.)

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© Boardworks Ltd of 60 Finding the distance between two points We can use Pythagoras’ Theorem to find the distance between any two points on a co-ordinate grid. Find the distance between the points A(7, 6) and B(3, –2). We can add a third point at C to form a right-angled triangle. AB 2 = AC 2 + BC 2 = = = 80 AC = 80 = 8.94 units (to 2 d.p.) A(7, 6) B(3, –2) C (7, –2)

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© Boardworks Ltd of 60 Finding the distance between two points What is the distance between two points with coordinates A( x A, y A ) and B( x B, y B )? The horizontal distance between the points is x B – x A The vertical distance between the points is y B – y A Using Pythagoras’ Theorem the square of the distance between the points A( x A, y A ) and B( x B, y B ) is ( x B – x A ) 2 + ( y B – y A ) 2 The distance between the points A( x A, y A ) and B( x B, y B ) is ( x B – x A ) 2 + ( y B – y A ) 2

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© Boardworks Ltd of 60 Solving problems using Pythagoras’ Theorem As we have seen Pythagoras’ Theorem can be used to solve a great many problems involving the lengths in right-angled triangles. When solving a problem using Pythagoras’ Theorem always do the following: State Pythagoras’ Theorem for the given side lengths before substituting any values. Draw a clearly labeled diagram of the situation, making sure you identify which side is the hypotenuse and which are the shorter sides. Make sure that your answer is sensible given the lengths of the other two sides.

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© Boardworks Ltd of 60 A A A A A A S2.6 Applying Pythagoras’ Theorem in 3-D Contents S2 Pythagoras’ Theorem S2.3 Pythagorean triples S2.1 Introducing Pythagoras’ Theorem S2.2 Identifying right-angled triangles S2.4 Finding unknown lengths S2.5 Applying Pythagoras’ Theorem in 2-D

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© Boardworks Ltd of 60 The longest diagonal in a cuboid Pythagoras’ Theorem can also be applied to three-dimensional problems. For example, what is the length of the longest diagonal in a cuboid measuring 5 cm by 7 cm by 8 cm? 7 cm 5 cm 8 cm Start by labeling the vertices of the cuboid. A B C D E F G H The longest diagonal is CE. CE is the hypotenuse of right-angled triangle ACE. We could also use AG, BH or DF.

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© Boardworks Ltd of 60 The longest diagonal in a cuboid Pythagoras’ Theorem can also be applied to three-dimensional problems. For example, what is the length of the longest diagonal in a cuboid measuring 5 cm by 7 cm by 8 cm? Before we can find the length of CE, we need to find the length of AC. 7 cm 5 cm 8 cm A B C D E F G H We can do this by applying Pythagoras’ Theorem to triangle ABC. AC 2 = AC 2 = AC 2 = 74

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© Boardworks Ltd of 60 The longest diagonal in a cuboid Pythagoras’ Theorem can also be applied to three-dimensional problems. For example, what is the length of the longest diagonal in a cuboid measuring 5 cm by 7 cm by 8 cm? 7 cm 5 cm 8 cm Now we know that AC 2 = 74, we can use Pythagoras’ Theorem to work out the length of CE. A B C D E F G H CE 2 = AE 2 + AC 2 CE 2 = CE 2 = CE 2 = 138 CE = cm (to 2 d.p.)

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© Boardworks Ltd of 60 The longest diagonal in a cuboid Let’s call the diagonal across the base e. d a b c e Using Pythagoras’ Theorem, e 2 = a 2 + b 2 and d 2 = e 2 + c 2 So, d 2 = a 2 + b 2 + c 2 Show that the longest diagonal d in a cuboid measuring a by b by c is given by the formula d = ( a 2 + b 2 + c 2 )

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© Boardworks Ltd of 60 The longest diagonal in a cuboid Substituting the values into the formula, we have A cuboid has sides of length 8 cm, 10 cm and 11 cm. Use the formula d = ( a 2 + b 2 + c 2 ) to find the length of the longest diagonal d to 2 d.p. d = ( ) = ( ) = 285 = cm (to 2 d.p.)

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