A Math Forum Project

Geometry Forum - Problem of the Week
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    Lindsey Becker

    ____________________

    It is possible for Jill to form a right angle to the dock and the bridge
if she was traveling in a semicircle.  The diameter of the semicircle is
the segment joining the dock to the end if the bridge.  Jill can be at
any given point on the semicircle and form an inscribed right angle.
Therefore, the biconditional that says an inscribed angle is a right
angle if and only if it is inscribed in a semicircle proves Jill's lines
of sight are perpendicular if she travels in a semicircle.

    Matt Bouton

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    Jill followed a path such that her lines of sight to the two points, dock 
and bridge, were always perpendicular.  Lines form an angle, angles 
inscribed in a semicircle are always 90 degrees; meaning that, if her 
lines of sight were always perpendicular to each other, she must 
have traveled in a semicircle.

    Bob Toboz, Amy Varga, Tami Trimmer, and Becky Grimm

    ____________________

    Jill had to have traveled in a semicircle down river.  If she would 
have traveled in any other pattern, her path would nave not been 
perpendicular from her sight.

    Jill Grobelski and Derek Morrison

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    We concluded that you can put Jill anywhere on a semicircle, because 
an angle inscribed in a semicircle is a right angle.  So, as long as she 
stayed on the arc, she would be at a 90 degree angle.

    Ryan Ferchak and Bob Gallagher

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    Corollary 3 (in our Geometry book) states that an angle inscribed in a
semicircle is a right angle.  Therefore she traveled in a semicircular path.

    Daniel Chan

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         The following solution came from Daniel Chan, a Grade 10 student 
at Burnaby South 2000 Secondary in Burnaby, BC.  I found it 
particularly interesting because his solution constitutes a proof of the 
angle in a semi-circle theorem that I have never seen before!
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Jill travels in a circle path with the midpoint of the dock and the foot 
of the bridge as its centre.

Let O be the mid point of the foot of the bridge (B) and the dock (D)
Let J be where Jill is.
Make the rectangle JBCD.

                   B --------------------------J
                   |  .                   .    |
                   |       .         .         |
                   |            .              |
                   |       .    O    .         |
                   |  .                    .   |
                   C---------------------------D

To Prove:  OJ is still the same when DJ and BJ change.
Proof:     In JBCD, BD is the diagonal.
           therefore,
           OJ is half the diagonal with O as the intersection
           of the two diagonals of JBCD
           BD will not change even when DJ and BJ change because BD is    
           the distance between the dock and foot of the bridge.
           OJ = BD/2
           So OJ also not change even DJ and BJ change.
           With OJ not change, its distance and point O is static.
           So Jill would travel a circle path with O its centre and OJ
           the radius of the circle.

    Don Kunc

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    The path in which Jill traveled was a semi-circle.  The way that I  
came up with this conclusion was I drew an infinite number of lines  
that are perpendicular to the to both the dock and the foot of the  
bridge.  The way I drew the segments was, I took the corner of a  
piece of paper, knowing that the intersection is a right angle, I  
drew the lines on a drawing of the bridge and the dock.  After I drew  
about 20 of the perpendicular lines, I figured out a pattern.  When  
you connect the points of intersection of all the perpendicular  
lines, the figure you come out with is a semi-circle.  Thus the pathe  
she traveled in is a semi-circle.

    Chris Taormina

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            If you draw a line from the dock to the bridge, then this  
becomes the diameter of a half circle.  Also from this segment you  
make an isoscles right triangle with the right angle opposite the  
diameter.  Taking the median of this triangle you get that half of  
the diameter is congruent to the median.  In other words they are  
radii of the half circle.  This gives us that Jill will be one radius  
away from the midpoint of the diameter throughout her whole trip.   
Also since taking the endpoints of the diameter and any other point  
of a circle and connecting them all, you will get a right angle  
opposite the diameter.  With both of these ideas you can prove that  
Jill was traveling in an arc on the water.
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30 June 1995