Pythagoras' Theorem Tilted Squares - Areas of Squares Area = 1cm^2 4cm^2 9cm^2 16cm^2 Area = 2cm^2 5cm^2 10cm^2 17cm^2 Area of Tilted Squares Formula: Area of a Square c c A = c^2 Formula: Area of a Square with a tilt of 1 A = c^2 + 1 c c WHAT? 1. 2. 3. 4. Calculated the area of the tilted squares using the Pythagoras Theorem Pythagoras Theorem: a^2 + b^2 = c^2 HOW? The sum of the areas of squares a and b equals the area of square c. Square c represents the length of the hypotenuse, which is the longest side of a right-angled triangle and is opposite the right angle.

a and b represent the lengths of the other two sides, also known as legs :. 3. a^2 + b^2 = c^2

1^2 + 3^2 = 10

√10 = 3.16

c = 3.16 cm : Pythagorean Theorem PROOF a^2 + b^2 = c^2 Area of entire square: (a+b)(a+b)

Area of small square: c^2

Area of triangle: 1/2ab

4 triangles: 4(1/2ab)

:. Small square + Triangles = c^2 + 2ab = (a+b)(a+b)

:. c^2 + 2ab = a^2 + 2ab + b^2

:. c^2 = a^2 + b^2 3cm :. Formula: Area of a Square with a tilt of n Areas of Squares with a tilt of n Notice that the areas increase in a certain pattern - adding odd numbers

2cm^2 (+3) = 5cm^2 (+5) = 10cm^2 (+7) = 17cm^2 (+9) = 26cm^2 (+11) = 37cm^2 a^2 + b^2 = c^2

2^2 + 3^2 = 13

√13 = 3.61 = c

A = 13cm^2

13 - 9 =4 a^2 + b^2 = c^2

3^2 + 3^2 = 18

18 = 4.24 = c

A = 18cm^2

18 - 9 = 9 Tilt of 2 Tilt of 3 Conclusion Original Square c^2 = 3^2 = 9cm^2 :. Formula = c^2 + 4 :. Formula = c^2 + 9 A = c^2 + n^2 Tilt of 1 is c^2 + 1

Tilt of 2 is c^2 + 4

Tilt of 3 is c^2 + 9 n is always a perfect square Formulae The area of a tilted square can be easily found using the formula: c^2 + n^2 Where c^2 is the area of a normal square + n^2 which is the square of the tilt = the area of the tilted square a, b & c are Pythagorean triples - 3 positive integers

Pythagorean triples describe the three integer lengths of a right-angled triangle

:. right-angled triangles with non-integer sides don't form Pythagorean triples. The Pythagorean Theorem is a theorem by Pythagoras that the square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the other two legs. 1cm 3cm 3cm a b a b Square root of Square root of Square root of By Michelle, Ben & Snigdha

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