document.write( "Question 1011854: A ladder that is 6m long rests with one end on the horizontal ground and its other end against a vertical wall. Considering the ground and the wall as the x and y axes respectively, find the locus of the possible midpoints of the ladder. \n" ); document.write( "

Algebra.Com's Answer #627654 by ikleyn(52803) $\"\"$ $\"About$

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\n" ); document.write( "A ladder that is 6m long rests with one end on the horizontal ground and its other end against a vertical wall. Considering the ground and the wall as the x and y axes respectively, find the locus of the possible midpoints of the ladder.
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document.write( "Answer\r\n" );
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document.write( "The locus is the arc of a circle, namely, a 90° arc.\r\n" );
document.write( "The center of the circle is at the base of the wall.\r\n" );
document.write( "The radius of the circle is 3m, which is half of the length of the ladder.\r\n" );
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document.write( "Proof\r\n" );
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document.write( "Let us connect the midpoint of the ladder with the base of the wall (I mean: let us draw this imaginary line . . . ).\r\n" );
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document.write( "This line is the median in our right-angled triangle.\r\n" );
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document.write( "It is well known fact that in an right-angled triangle the median drawn to the hypotenuse has the length \r\n" );
document.write( "half of the hypotenuse (see the lesson Median drawn to the hypotenuse of a right triangle in this site).\r\n" );
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document.write( "Thus the midpoint of the ladder is always at the distance half of the ladder length from the base of the wall.\r\n" );
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document.write( "So the locus is the arc of the circle.\r\n" );
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document.write( "To estimate the angle of the arc, consider two extreme positions of the ladder: horizontal and vertical.\r\n" );
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