The Pythagorean Theorem

Pre-Algebra · Axiom Academy

LESSON The Pythagorean Theorem In every right triangle, the two legs and the longest side lock together by one rule: a^2 + b^2 = c^2 . 1. The Sides of a Right Triangle A right triangle has exactly one right angle — the little square corner. The two sides that form that corner are the legs , a and b . The side directly across from the right angle is the hypotenuse , c , and it is always the longest side. Those three lengths are never independent — they obey the Pythagorean theorem. One of the two sides that meet at the right angle. Here a = 3 . The other side at the right angle. Here b = 4 . Opposite the right angle, and always the longest side. Here c = 5 . The square corner is what makes the rule work. Remove it and a^2 + b^2 = c^2 no longer holds. Square each leg, add them, and you get the hypotenuse squared A right triangle with legs 3 and 4 has hypotenuse 5 2. Why It Is True: Squares on the Sides The little 2 in a^2 means a square . So build a real square on each side of the right triangle: the square on leg a has area a^2 , the square on leg b has area b^2 , and the square on the hypotenuse has area c^2 . The Pythagorean theorem says the two smaller squares' areas add up to the big square's area. The 9 tiles and the 16 tiles exactly fill the 25-tile square Because a^2 + b^2 = c^2 links all three sides, knowing any two of them lets you find the third. There are two cases. Legs 6 and 8 give a hypotenuse of 10. Worked leg: hypotenuse 13, leg 5

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