document.write( "Question 407117: What is the proof for Theorem 7.6 Geometric Mean (Altitude): In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments \n" ); document.write( "

Algebra.Com's Answer #287043 by richard1234(7193) $\"\"$ $\"About$

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Yes, the altitude divides the hypotenuse into two segments:\r
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\n" ); document.write( "\n" ); document.write( "I think you want to show that the geometric mean of the two segments equals the length of the altitude, which is a famous theorem.\r
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\n" ); document.write( "\n" ); document.write( "The proof comes from the fact that triangles ABD and CBD are similar, which I've already shown. Therefore, we can let $\"AB+=+x\"$ , $\"BD+=+xy\"$ , and $\"DC+=+xy%5E2\"$ . The geometric mean of AB and DC is given by\r
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\n" ); document.write( "\n" ); document.write( " $\"sqrt%28x%2A%28xy%5E2%29%29+=+xy\"$ which is the same as BD, so we're done. \n" ); document.write( "

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