document.write( "Question 1207909: Prove that if two nonvertical lines have slopes whose product is -1, then the lines are perpendicular.
\n" ); document.write( " \n" ); document.write( "

Algebra.Com's Answer #846043 by ikleyn(52781) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "Prove that if two nonvertical lines have slopes whose product is -1, then the lines are perpendicular.
\n" ); document.write( "~~~~~~~~~~~~~~~~~~~\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "The slope of a straight line is a value of the tangent function of the angle between this line and x-axis.\r\n" );
document.write( "\r\n" );
document.write( "So, we can write   = ,   = ,  where    and    are the angles \r\n" );
document.write( "that these straight line form with x-axis.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The angle between these straight lines is  .  Let's calculate the tangent of this angle\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "     = .\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The denominator    is   = 1 + (-1) = 0,  so    is not defined.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "But the tangent function of an angle is not defined if and only if the angle is +/- 90 degrees.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Thus we proved that the angle between our lines is +/- 90 degrees.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "It means that the lines are perpendicular.\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( "At this point, the proof is complete and the problem is solved.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );