Sec. 8-1 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 2 + b 2 = c 2

Remember  Hypotenuse  Side across from the right angle  Always the longest side  It is the c in Pyth. Thm.  Legs of a right triangle  Sides that form the right angle  Does not matter which you call a and b

Examples a)Right triangle with sides 20, 29, & 21. Which is hypotenuse and verify Pyth. Thm. b)Find distance from A to B c)Find hypotenuse if legs are 7 & 24 A B 25 35

Pythagorean Triple  Set of whole numbers that satisfy the Pythagorean Theorem

Common Pythagorean Triples 33, 4, 5 55, 12, 13 88, 15, 17 77, 24, 25 AAlso any multiples of these are Pythagorean triples

Find the length of the hypotenuse of the right triangle. Tell whether the side lengths form a Pythagorean triple. Finding the Length of a Hypotenuse Because the side lengths 5, 12, and 13 are whole numbers, they form a Pythagorean triple. 12 x 5 S OLUTION c 2 = a 2 + b 2 Pythagorean Theorem Substitute. x 2 = Square. x 2 = Add. x 2 = 169 Find the square root. x = 13

Sometimes the answer needs to be exact.  This means you must leave the answer in simplest radical form.

Finding the Length of a Leg Many right triangles have side lengths that do not form a Pythagorean triple. Find the length of the leg of the right triangle. c 2 = a 2 + b 2 Pythagorean Theorem Find the square root. 147 = x Subtract 49 from each side. 147 = x 2 Square. 196 = 49 + x 2 Substitute.14 2 = x 2 x 14 7 Use product property = x Simplify the radical. 7 3 = x S OLUTION

Ex. 2  Leave in simplest radical form 20 8 x

Finding Area 12 m 20 m h

Another Example Find the area  53 cm h 7 cm

Converse of 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 2 sides, then it is a right triangle If a 2 + b 2 = c 2, then the triangle is right

Ex. 5 Is it a right triangle?

If c 2 > a 2 + b 2, then the triangle is obtuse If c 2 < a 2 + b 2, then the triangle is acute Remember you are comparing the hypotenuse.

Classify Ex. 6 a)6, 11, 14 b)12, 13, 15

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