Problem 1 Finding the Length of the Hypotenuse

What is the length of the hypotenuse of Δ A B C ? cap delta eh b c , question mark Do the side lengths of Δ A B C cap delta eh b c form a Pythagorean triple? Explain.

( leg 1 ) 2 + ( leg 2 ) 2 = ( hypotenuse ) 2 Pythagorean Theorem a 2 + b 2 = c 2   21 2 + 20 2 = c 2 Substitute  21  for  a  and  20  for  b . 441 + 400 = c 2 Simplify . 841 = c 2   c = 29 Take the positive square root . table with 6 rows and 3 columns , row1 column 1 , open , leg sub 1 , close squared . plus . open , leg sub 2 , close squared , column 2 equals . open . hypotenuse . close squared , column 3 cap pythagoreancap theorem , row2 column 1 , eh squared , plus , b squared , column 2 equals , c squared , column 3 , row3 column 1 , 21 squared , plus , 20 squared , column 2 equals , c squared , column 3 cap substitute . 21 , for , eh , and , 20 , for , b . , row4 column 1 , 441 plus 400 , column 2 equals , c squared , column 3 cap simplify , . , row5 column 1 , 841 , column 2 equals , c squared , column 3 , row6 column 1 , c , column 2 equals 29 , column 3 cap takethepositivesquareroot . . , end table

The length of the hypotenuse is 29. The side lengths 20, 21, and 29 form a Pythagorean triple because they are whole numbers that satisfy a 2 + b 2 = c 2 . eh squared , plus , b squared , equals , c squared , .

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1. 1. The legs of a right triangle have lengths 10 and 24. What is the length of the hypotenuse?
   2. Do the side lengths in part (a) form a Pythagorean triple? Explain.

Problem 2 Finding the Length of a Leg

Algebra What is the value of x? Express your answer in simplest radical form.

a 2 + b 2 = c 2 Pythagorean Theorem 8 2 + x 2 = 20 2 Substitute . 64 + x 2 = 400 Simplify . x 2 = 336 Subtract  64  from each side . x = 336 Take the positive square root . x = 16 ( 21 ) Factor out a perfect square . x = 4 21 Simplify .       table with 8 rows and 3 columns , row1 column 1 , eh squared , plus , b squared , column 2 equals , c squared , column 3 cap pythagoreancap theorem , row2 column 1 , 8 squared , plus , x squared , column 2 equals , 20 squared , column 3 cap substitute . . , row3 column 1 , 64 plus , x squared , column 2 equals 400 , column 3 cap simplify , . , row4 column 1 , x squared , column 2 equals 336 , column 3 cap subtract . 64 . fromeachside . . , row5 column 1 , x , column 2 equals , square root of 336 , column 3 cap takethepositivesquareroot . . , row6 column 1 , x , column 2 equals . square root of 16 open 21 close end root , column 3 cap factoroutaperfectsquare . . , row7 column 1 , x , column 2 equals 4 square root of 21 , column 3 cap simplify , . , row8 column 1 , , column 2 , column 3 , end table

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2. The hypotenuse of a right triangle has length 12. One leg has length 6. What is the length of the other leg? Express your answer in simplest radical form.