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Phythagoras Solving a missing side of a right angled triangle. Miss Hudson’s Maths

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Objective for the next week and a half - To solve problems involving the Pythagorean Theorem. For Right Angled Triangles Only! Hypotenuse - always opposite the right angle Miss Hudson’s Maths

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Step one - You must always identify which side is the hypotenuse in your given triangle. leg hypotenuse a b c This helps us to identify which sides are where in our formula. Miss Hudson’s Maths

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ba ab Hypotenuse c Pythagorean Theorem In our formula c always represents the longest side known as the hypotenuse and a and b always represent the shorter sides. Miss Hudson’s Maths

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However each right angled triangle can have different letters to identify the sides. This doesn’t change the formula that we use! We must still identify the hypotenuse and then the short sides like normal. B S T Miss Hudson’s Maths

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6 8 x Solve for x. Miss Hudson’s Maths

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4 7 y Solve for y. Miss Hudson’s Maths

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To the nearest meter, find the length of the diagonal of a rectangle with a width of 4 meters and a length of 10 meters. 4 m. 10 m. x Miss Hudson’s Maths

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20 kms A car drives 20 kilometres due east and then 45 kilometres due south. To the nearest kilometre, how far is the car from its starting point? 45 kms x Miss Hudson’s Maths

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Written Questions: A ladder is leaning against a wall. The base of the ladder is 9m away from the base of the wall and the top of the ladder reaches 16m up the wall. What is the length of the ladder? Hint: Draw a diagram to help your understanding. Give the unknown length of the ladder a pronumeral. Miss Hudson’s Maths

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Written Questions: On my way to school I walk around a sporting complex. I travel 2.4kms east and then turn south and walk another 3kms. On the way home I decide to take a shortcut through the sporting complex and walk directly home. What distance will I walk when I travel home? Miss Hudson’s Maths

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Written Questions: In the local Darch park a massive tree has been damaged by a storm. The Wanneroo shire decides they need to stabilize the tree by attaching wires to the top of the tree and the ground. If the tree is 17 metres tall and the shire workers want to connect the wires 7.5 meters from the base of the tree how much wire will they require? Miss Hudson’s Maths

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6 t 15 Solve for t. Miss Hudson’s Maths

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Mixed Problems” - missing the hypotenuse or short side. Miss Hudson’s Maths

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12 cm 5 cm c c 2 = 5 2 + 12 2 c 2 = 169 c =√169 c = 13 cm Eg 1: When calculating the length of the hypotenuse, we ADD the squares of the other 2 sides. Eg 2: When calculating the length of one of the shorter two sides we SUBTRACT the squares. 6 cm 3 cm x x 2 = 6 2 - 3 2 x 2 = 27 x = √27 x = 5.2 cm (1dp) Miss Hudson’s Maths

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Applications A ladder is placed up the side of a building. The ladder is 8 m long, and the foot of the ladder is 3 m out from the wall. Draw a diagram to illustrate this, and calculate how far up the building the ladder reaches. Let x be the height up the building. x 2 = 8 2 - 3 2 x 2 = 55 x = √55 x = 7.42 (2dp) The ladder reaches 7.42 m up the side of the building. Miss Hudson’s Maths

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A square has a length of 11mm. Find the length of the diagonal of the square. 11mm s Miss Hudson’s Maths

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Find the width of a rectangle with a length of 20cm and a diagonal of 24cm. 20cm w 24cm Miss Hudson’s Maths

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A rectangle has a length of 14cm and a width of 8cm. Find the length of the diagonals of this rectangle. 14cm 8cm d Miss Hudson’s Maths

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Extension Question Find the length of a square which has a diagonal of 16cm. Note: Both of the short sides will be worth the same value because it is a square. Miss Hudson’s Maths

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Extension Question Find the length of a square which has a diagonal of 27cm. Note: Both of the short sides will be worth the same value because it is a square. Miss Hudson’s Maths

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Extension Question Find the slant height of a triangle if the area is 96m 2 and has a base of 16m. Miss Hudson’s Maths

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Extension Question Find the slant height of a triangle if the area is 75m 2 and has a base of 15m. Miss Hudson’s Maths

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Pythagorean Triples 3 4 5 6 8 10 9 12 15 12 16 20 Miss Hudson’s Maths

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Pythagorean Triples 3 4 5 6 8 10 9 12 15 12 16 20 5 12 13 10 24 26 15 36 39 7 24 25 14 48 50 21 72 75 9 15 12 Miss Hudson’s Maths

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Informal Proof #1 Inscribe a square within the square. Miss Hudson’s Maths

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a b a a a b b b c c c c Informal Proof #1 Miss Hudson’s Maths

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a b a a a b b b c c c c a b c Informal Proof #1 Miss Hudson’s Maths

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b a a a b b c c c a b c Informal Proof #1 Miss Hudson’s Maths

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b a a a b b c c c a b c a b c Informal Proof #1 Miss Hudson’s Maths

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b a a b c c a b c a b c Informal Proof #1 Miss Hudson’s Maths

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b a a b c c a b c a b c b a c Informal Proof #1 Miss Hudson’s Maths

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a b c a b c a b c b a c Informal Proof #1 Miss Hudson’s Maths

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a b c a b c a b c b a c a b c Informal Proof #1 Miss Hudson’s Maths

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a b c a b c b a c a b c Informal Proof #1 Miss Hudson’s Maths

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a b c a b c b a c a b c Informal Proof #1 Miss Hudson’s Maths

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a b c a b c b a c a b c Informal Proof #1 a b c Miss Hudson’s Maths

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a b c b a c a b c Informal Proof #1 a b c a b c Miss Hudson’s Maths

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b a c a b c Informal Proof #1 a b c a b c a b c Miss Hudson’s Maths

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b a c Informal Proof #1 a b c a b c a b c b a c Miss Hudson’s Maths

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Informal Proof #1 a b c a b c a b c b a c Miss Hudson’s Maths

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a b a a a b b b c c c c Informal Proof #2 a + b Total Purple Yellow Area Area Area - = - = Miss Hudson’s Maths

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