Geometry Forum - Problem of the Week
Solutions - North Star and Latitude, March 7-11, 1994
Jim Sadowski
____________________
I have proved the lady sailor's solution to be right. I
projected the earth's sphere onto a flat plane to form a circle. In
the problem there are two angles: The latitude angle (1) which is the
angle between the equator plane and the segment from the
center of the earth to the location of the boat. The other angle is the
"North Star" angle (2) which is the angle between the horizon plane
and the segment from the location of boat to the North Star.
The axis of the earth and the equator are perpendicular in a
plane. The segment from the North Star (NS) to the boat location is
parallel to the earth's axis. The segment from the center of the
earth to the boat location (CE) is perpendicular to the horizon
plane. I proved they were perpendicular because the horizon line is
tangent to the circle at the boat location. CE is the radius of the
circle and by a theorem in our book, a radius of a circle to the point
of tangency is always perpendicular to the tangent line. CE acts as
the transversal of the parallel lines NS and the earth's axis. Angle 1
and 2 are alternate interior angles of the transversal. Alternate
interior angles are always congruent by the PAI theorem in our
book. But angle 1 and 2 are cut into two different angles. I proved
one half of one angle congruent to the other half because they are
both ninety degree angles by definition of perpendicular. That makes
the other half of angles 1 and 2 congruent. Angle 1 is latitude
measure so angle 2 is measure of the angle of North Star. This makes
the lady sailor's assumption correct.
Gino Perrotte
____________________
The imaginary axis through the center of the Earth is parallel to the
view the companion sailor has to the North Star. This is found to
be true because the North Star is in the Northern sky. The center of
the Earth is used as a reference point. Using an imaginary compass,
place the hinge on the center of the Earth. The side of the compass
with the point is to be lined up on the equator. Place the pencil point
where the boat is. This line is the latitude that you are on. The side
of the compass that is opened with the pencil point is a transversal of
the two parallel lines. The two parallel lines are the Axis and the
view from the sailor. Since the lines are parallel and there is a
transversal, then the latitude measure is congruent to the view
measure by alternate interior angles.
The companion is correct.
pencil point side
+
Transversal +
+
hinge +
---+++++++++-------------
side aligned with the equator
Nick High
____________________
According to Webster's Dictionary, the term latitude is
defined as an angular distance north or south from the earth's
equator measured through 90 degrees. When at the north pole, the
North star would be directly above, forming a 90 degree angle with
the horizon. When at the equator, the North star would be on the
horizon and barely seen, forming a zero degree angle with the
horizon. Using the definition of latitude and the information
already reported, the degree latitude would equal the degree of the
angle measurement. Assuming that the companion who said, "That
angle is equal to our latitude" knew the latitude that the two were at,
then the comment she made was correct.
Valerie Stalter
____________________
The latitudes of the earth are formed from the angles with one ray being the equator. The angle from the North Star(A) would be from the horizon to the star. If you'd extend line SA, to the equator, the angle formed with the equator(a) would be congruent to the angle(A). To prove this make a line parallel to the equator at the pole. Sa would then be a transversal making A congruent to a by the PAI theorem. Since angle a is congruent to the angle formed by the star, angle A would be the latitude.
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