The Pythagorean Theorem and Its Converse

Pythagorean Theorem In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. a b c a2 + b2 = c2

Pythagoras was a Greek mathematician who lived in the 6th century B.C. http://www.arcytech.org/java/pythagoras/history.html

The Babylonians, Egyptians, and Chinese knew about this relationship in a right triangle, but Pythagoras made the relationship popular and related it to geometry. The Egyptians used a rope with 12 evenly spaced knots to form perfect corners for buildings and pyramids. http://www.arcytech.org/java/pythagoras/history.html

“Pythagoras also headed a cult known as the secret brotherhood that worshiped numbers and numerical relationships. They attempted to find mathematical explanations for music, the gods, the cosmos, etc. Pythagoras believed that all relations could be reduced to number relations.” http://www.arcytech.org/java/pythagoras/history.html

A Pythagorean triple is a set of nonzero whole numbers that “work” in the Pythagorean theorem: 3, 4, 5 5, 12, 13 8, 15, 17 7, 24, 25 3 4 5 5 12 13

A Pythagorean family is formed by multiples of a triple. 3, 4, 5 6, 8, 10 (multiply by 2) 9, 12, 15 (multiply by 3) 30, 40, 50 (multiply by 10)

Find the value of the variable Find the value of the variable. Leave your answer in simplest radical form.

Find the area of the triangle.

Converse of the Pythagorean Theorem If the square of the length of one side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle.

If the square of the length of the longest side of a triangle is greater than the sum of the squares of the lengths of the other two sides, then the triangle is obtuse: If c2 > a2 + b2 , the triangle is obtuse.

Theorem 7-7 If the square of the length of the longest side of a triangle is less than the sum of the squares of the lengths of the other two sides, then the triangle is acute: If c2 < a2 + b2 , the triangle is acute.

The Scarecrow’s Theorem (Wizard of Oz) "The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side." Is this correct?

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If life was a video game how would you play it when things got hard If life was a video game how would you play it when things got hard? Explain how your answer is relevant to a video game at least 3 sentences.