Section 8 6 The Pythagorean Theorem and Radical

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Section 8. 6 The Pythagorean Theorem and Radical Equations Copyright © 2012, 2009, 2005,

Section 8. 6 The Pythagorean Theorem and Radical Equations Copyright © 2012, 2009, 2005, 2002 Pearson Education, Inc.

Pythagorean Theorem In any right triangle, if c is the length of the hypotenuse

Pythagorean Theorem In any right triangle, if c is the length of the hypotenuse and a and b are the lengths of the two legs, then c 2 = a 2 + b 2. hypotenuse c b legs a Copyright © 2012, 2009, 2005, 2002 Pearson Education, Inc. 2

Example The ramp to Tony Pitkin’s barn rises 5 feet over a horizontal distance

Example The ramp to Tony Pitkin’s barn rises 5 feet over a horizontal distance of 12 feet. How long is the ramp? That is, find the length of the hypotenuse of a right triangle whose legs are 5 feet and 12 feet. ? 5 12 The hypotenuse is 13 feet. Length cannot be negative. The solution will check. Copyright © 2012, 2009, 2005, 2002 Pearson Education, Inc. 3

Example A baseball diamond is square. Each side of the square is 90 feet

Example A baseball diamond is square. Each side of the square is 90 feet long. How far is it from home plate to second base? ? 90 90 The distance is approximately 127. 3 feet. Again the length cannot be negative. Copyright © 2012, 2009, 2005, 2002 Pearson Education, Inc. 4

Radical Equations A radical equation is an equation with a variable in one or

Radical Equations A radical equation is an equation with a variable in one or more of the radicands. It can be simplified by squaring both sides of the equation. Copyright © 2012, 2009, 2005, 2002 Pearson Education, Inc. 5

Example Solve and check. Square each side. Simplify and solve for x. Check: Copyright

Example Solve and check. Square each side. Simplify and solve for x. Check: Copyright © 2012, 2009, 2005, 2002 Pearson Education, Inc. 6

Example Solve and check. Square each side. Simplify and solve for x. Check: False

Example Solve and check. Square each side. Simplify and solve for x. Check: False There is no solution. Copyright © 2012, 2009, 2005, 2002 Pearson Education, Inc. 7

Example Solve and check. Isolate the radical. Square each side. FOIL each side. Simplify

Example Solve and check. Isolate the radical. Square each side. FOIL each side. Simplify and set equal to 0. Factor the quadratic equation. Solve for y. Copyright © 2012, 2009, 2005, 2002 Pearson Education, Inc. 8

Example (cont) Solve and check. Check: False! 4 is an extraneous root. The only

Example (cont) Solve and check. Check: False! 4 is an extraneous root. The only solution is y = 0. Copyright © 2012, 2009, 2005, 2002 Pearson Education, Inc. 9

Example Solve and check. Check: The solution is 2. Copyright © 2012, 2009, 2005,

Example Solve and check. Check: The solution is 2. Copyright © 2012, 2009, 2005, 2002 Pearson Education, Inc. 10